'  GIFT  OF 
«!•  0.  Hirschf elder 


HEAT 


HEAT 


BY 


MARK    R.    WRIGHT 

AUTHOR   OF 
SOUND,    LIGHT,    AND   HEAT,"    AND   "ELEMENTARY  PHYSICS ' 


LONGMANS,    GREEN,    &    CO. 

NEW   YORK:    15  EAST  l6th  STREET 
'893 

All  rights  reserved 


2- 


PREFACE 

THE  present  work  is  intended  for  those  who  have  read  the 
elementary  parts  of  the  subject,  as  treated  in  "  Sound,  Light, 
and  Heat,"  or  who  are  able  at  once  to  attack  a  more  advanced 
work.  It  is  an  attempt  to  place  the  leading  facts  and  prin- 
ciples of  the  subject  before  the  student.  The  point  of  view 
of  the  writer  is  that  of  a  teacher,  and  this  will  explain  why 
certain  parts  are  treated  more  elaborately  than  others,  and 
why  worked  numerical  examples  occur  so  frequently  in  the 
text  and  at  the  end  of  each  chapter.  Such  examples  should 
not  be  merely  copied  ;  they  are  inserted  because  it  is  found 
that  a  carefully  worked  example  proves  of  great  use  to  the 
student,  who  naturally  finds  a  difficulty  in  understanding  the 
generalizations  of  a  science. 

A  fair  selection  of  the  experiments  described  should  be 
performed  by  the  student.  The  experiments  selected  will 
depend  upon  the  apparatus  available  and  the  time  at  his 
disposal ;  they  should  be  typical.  A  high  form  of  accuracy 
is  not  required  at  this  stage ;  it  will  be  sufficient  if  he  can 
clearly  indicate  the  minor  corrections  that  have  been  omitted, 
and  how  such  corrections  would  affect  the  results.  It  is 
believed  that  the  description  of  the  experiments  will  enable 
any  one  to  repeat  them  ;  to  have  gone  into  further  detail 
would  have  made  the  work  too  bulky,  and  many  details  can 
only  be  mastered  either  by  instruction  in  the  laboratory  or 
by  the  slow  yet  efficient  method  of  trial  and  failure.  Fortu- 
nately, the  once  prevalent  method  of  studying  a  science 


475450 


vi  Preface 

from  a  text-book  alone  is  rapidly  disappearing.  It  is  neces- 
sary, however,  to  point  out  that  too  much,  as  well  as  too  little, 
time  may  be  spent  over  experimental  science  ;  mental  inertia 
is  as  possible  in  the  laboratory  as  in  the  lecture-room. 

Lack  of  apparatus  and  skill  in  manipulation  will  prevent 
the  verification  of  some  experiments  ;  for  example,  Regnault's 
experiments  on  the  absolute  expansion  of  mercury  in  §  32 ;  but 
there  is  no  reason  why  the  method  should  not  be  thoroughly 
understood  after  the  simple  experiments  illustrated  in  Figs.  26 
and  27  have  been  made.  There  may  appear,  at  first  sight,  no 
reason  for  inserting  in  a  book  like  this  formulae  similar  to 
those  given  on  p.  43,  or  for  refinements  such  as  true  coefficient 
of  expansion  at  t°.  The  student  will,  however,  meet  with 
these  in  his  reading,  especially  if  he  refers,  as  it  is  trusted  he 
will  do,  to  more  elaborate  treatises  and  to  original  memoirs ; 
this  is  dealt  with  on  the  first  part  of  p.  46,  and  in  other  parts 
of  the  work. 

Reference  has  been  made  to  the  work  of  recent  experi- 
menters where  the  results  could  be  incorporated  in  a  book 
of  this  design.  Other  examples  of  recent  research  have  been 
omitted,  as  unsuitable  in  their  present  stage  of  development. 

The  permission  of  the  publishers  to  use  any  of  their  blocks 
has  made  the  task  of  illustrating  the  book  much  easier  than  it 
otherwise  would  have  been.  Some  cuts  have  been  utilized 
from  "  Sound,  Light,  and  Heat,"  and  thirty-five  have  been 
engraved  specially-. 

For  figures  and  tables,  reference  has  generally  been  made 
to  Everett's  "  Physical  Constants  ;  "  and  to  Jamin's  "  Cours  de 
Physique." 

§  74  is  inserted  so  that  the  student  may  revise  his  know- 
ledge of  dynamics,  and  to  lead  up  to  §  75,  which  was  written 
to  enable  the  reader  to  deal  readily  with  results  given  by 
different  experimenters  in  varying  units.  This  is  especially 
applicable  to  results  in  conductivity  (§  146). 

There  has  been  no  attempt  to  restrict  the  examples  or 
explanations  to  the  C.G.S.  units.  Such  a  restriction  may 
prove  useful  when  the  units  arc  universally  adopted,  but 
at  present  it  is  necessary  that  other  units  in  common  use 


Preface  vii 

should  be  understood,  and  the  change  from  one  set  to  another 
is  a  matter  of  easy  arithmetic. 

The  chapter  on  Thermo-Dynamics  is  of  an  elementary 
kind,  and  is  taken  up  with  an  attempt  to  explain  and  illustrate 
by  examples  the  first  two  laws  and  the  meaning  of  Joule's  and 
Thomson's  experiments.  The  published  papers1  of  both 
should  be  consulted  for  further  details ;  these  papers  will  be 
available  for  many  students. 

The  last  chapter  illustrates  many  portions  of  the  subject 
of  heat.  Professor  James  Thomson's  theory  of  atmospheric 
circulation  has  been  followed,  as  being  the  most  consistent 
of  the  theories  on  this  subject 

M.  R.  W. 


"Mathematical  and  Physical  Papers,"  by  Sir  Wm.  Thomson. 


CONTENTS 

CHAPTER  PAGE 

I.    HEAT— TEMPERATURE— THERMOMETERS i 

II.     EXPANSION  OF  SOLIDS 19 

III.  EXPANSION  OF  LIQUIDS 34 

IV.  DILATATION  OF  GASES 57 

V.    HEAT  AS  A  QUANTITY 79 

VI.     CHANGE  OF  STATE — THE  NATURE  OF  HEAT      ....  103 

VII.    CHANGE  OF  STATE — VAPORIZATION  AND  CONDENSATION  144 

VIII.    LIQUEFACTION  OF  GASES — VAPOUR-DENSITIES  ...  170 

IX.    HYGROMETRY 193 

X.    TRANSMISSION  OF  HEAT 207 

XI.    RADIATION     .     . 226 

XII.    THERMO-DYNAMICS 268 

XIII.    APPLICATIONS— CLIMATE 313 

EXAMINATION  PAPERS 330 

ANSWERS  TO  EXAMPLES 332 

INDEX 333 


H  EAT 

CHAPTER   I. 


ERRATA. 

Page    33,  Example  2,  read  coefficient  of  linear  expansion. 

„       33,  Example  6,  read  —^  =  — ,  and  100  divisions  of  this  scale  equal  18  inches. 
,,       78,  Example  7,  line  4,  for  of  read  inside. 
„      78.  Example  %,for  placed  in  read  forced  into. 
,,       87,  line  stjor  m'  read  m^. 
„     102,  line  i,  for  ball  read  vessel. 
„     102,  Example  t2,  read,  dry  air,  and  dry  hydrogen,  and  add,  at  normal  temperature 

and  pressure. 

,,     135,  line  10,  for  hyposulphate  read  hyposulphite. 
, ,     138,  line  6,  for  convex  read  concave. 
„     143,  Example  s,for  to  125  read  by  125. 

159,  line  9  from  bottom,  for  all  evaporated  read  finally  evaporated. 

160,  line  5,  read.'.  L  =  T  —  S  =  606*5  +  0-305^  -  /. 

168,  Example  5,  line  4,  read  of  dry  or  of  saturated  air. 

169,  line  4,  for  3-82  read  3-32. 

,  188,  line  19,  read  the  true  volume  is  60  X  (i  +  0-000025  X  30)  =  60  X  (i  +  0-00075). 

„  188,  line  21,  read .'.  4-21  should  be  divided  by  (i  +  0-00075}  =  4-21  -  0-0031. 

,,  188,  line  3  from  bottom,  readv  X  (i 

,,  206,  Example  4,  for  75-8  read  758. 

,,  206,  Example  %,for  7^58  read 758. 


ODjecr,  me  opinion  we.  crjme~TuwiT.ii  icspcct  tu  LUC  t^mp^xu. 
ture  not  only  depends  upon  the  temperature  of  the  object,  but 
also  upon  the  rate  at  which  heat  is  transferred  from  the  object 
to  the  hand,  or  from  our  hand  to  the  object.  The  transfer- 
ence of  heat  by  conduction  is  readily  illustrated  by  placing 


H  EAT. 


CHAPTER   I. 
HE  A  T— TEMPERA  TURE —  THERMOMETERS. 

1.  Temperature. — Terms  connected  with  the  subject  of 
heat  are  numerous  in  our  ordinary  language:  "hot,"  "cold," 
"  warm,"  and  others  suggest  themselves.  The  words  are  used 
to  describe  certain  sensations ;  the  agent  which  produces  the 
sensations  is  called  Heat.  The  word  we  use  to  describe  a 
particular  sensation  depends  upon  the  state  of  our  body  and 
upon  our  previous  experience.  In  winter  we  describe  a  day 
as  warm,  while  in  summer  we  should  describe  a  similar  day  as 
being  cold.  The  water  from  the  tap,  as  it  runs  over  our  hand, 
is  said  ordinarily  to  be  cold ;  if  the  hand  has  previously  been 
used  to  mix  ice  or  ice  and  salt,  the  water  feels  agreeably  warm. 
Not  only  does  our  opinion  vary  from  time  to  time,  but  each 
hand  may  lead  to  a  different  decision ;  for  example, ,  if  the 
right  hand  be  plunged  into  water  as  hot  as  we  can  bear  it,  and 
the  left  hand  be  plunged  into  melting  ice,  and  then  both  be 
placed  in  water  just  taken  from  the  tap,  the  right  hand  feels 
cold  and  the  left  hand  warm. 

The  state  a  body  is  in  with  respect  to  the  heat  that  affects 
the  senses  is  called  in  ordinary  language  its  Temperature. 

For  the  above  reasons  the  hands  cannot  be  used  for  the 
purpose  of  comparing  temperatures.  When  we  touch  an 
object,  the  opinion  we  come  to  with  respect  to  the  tempera- 
ture not  only  depends  upon  the  temperature  of  the  object,  but 
also  upon  the  rate  at  which  heat  is  transferred  from  the  object 
to  the  hand,  or  from  our  hand  to  the  object.  The  transfer- 
ence of  heat  by  conduction  is  readily  illustrated  by  placing 


v  ::-*:  •..•ffeat 

6fi  ..£&£.  lie'at  of  the  fire  travels  along  the 
poker  from  the  end  in  the  fire,  from  particle  to  particle,  until 
the  temperature  of  the  other  end  rises.  This  method  of 
transference  of  heat  is  called  Conduction. 

If  a  short  poker  and  a  piece  of  wood  of  equal  lengths  be 
placed  in  the  fire,  the  end  of  the  poker  out  of  the  fire  soon 
becomes  so  hot  that  we  dare  not  touch  it;  the  end  of  the 
piece  of  wood,  on  the  contrary,  can  easily  be  retained  in  the 
hand  ;  and  we  conclude  that  iron  is  a  good  conductor  of  heat, 
and  wood  a  poor  conductor;  wool,  felt,  etc.,  are  worse  con- 
ductors of  heat  than  wood. 

If  we  touch  pieces  of  wool  (the  carpet)  and  pieces  of 
metal  (fender,  etc.)  in  a  room  without  a  fire  and  shaded  from 
the  sun,  the  iron  feels  cold  and  the  wool  warm;  it  will 
be  shown  later  that  both  are  at  the  same  temperature.  The 
reason  for  the  different  effects  is  that  both  are  below  the 
temperature  of  the  hand.  Heat  flows  from  the  hand  to  the 
metal,  and  is  readily  conducted  to  other  parts ;  more  heat 
leaves  the  hand,  and  the  sensation  of  cold  is  the  result.  In 
the  case  of  the  wool,  the  heat  flows  from  the  hand  to  the 
carpet,  but  is  not  readily  conducted  to  other  parts  of  the 
carpet,  so  that  the  portion  touched  by  the  hand  soon  begins  to 
feel  warm.  If  the  metal  and  wool  were  above  the  temperature 
of  the  hand,  heat  would  flow  from  the  part  of  the  metal 
touched  to  the  hand,  and  from  other  parts  of  the  metal  to  the 
part  touched,  thence  to  the  hand,  and  the  metal  would  feel 
warm.  In  the  case  of  the  flannel  this  would  not  be  so  apparent ; 
heat  would  simply  be  transferred  from  the  part  touched  to  the 
hand,  but  it  would  flow  very  slowly  from  the  other  parts  of  the 
flannel  to  the  particular  part  touched,  and  thus  the  flannel 
would  not  feel  as  hot  as  the  metal. 

The  difficulty  of  comparing  temperatures  is  not  so  great 
when  we  deal  with  objects  all  made  of  the  same  substance. 
Bath  attendants  can  determine  with  great  accuracy  the  suitable 
temperature  for  a  bath,  and  we  can  readily  arrange  pieces  of 
silver  or  pieces  of  copper  according  to  their  temperature. 

The  temperature  does  not  depend  upon  the  amount  of  heat. 
If  we  take  a  pint  of  water  from  a  cistern  of  water,  both  are  at 


Heat— Temperature— Thermometers  3 

the  same  temperature,  and  yet  the  pint  contains  less  heat  than 
that  left  in  the  cistern.  An  analogous  state  of  things  exists  in 
the  terms  "  level "  and  "  quantity  "  of  water. 

If  two  vessels,  A  and  B  (Fig.  x),  containing  any  liquid,  be 
connected  by  a  tube,  the  flow  is  determined,  not  by  the  amount 
of  water  in  A  and  B,  but  by  the 
respective  levels ;  if  the  level 
of  the  water  in  A  be  higher 
than  that  in  B,  water  flows  from 
A  to  B ;  if  the  level  in  B  be 
higher  than  in  A,  the  flow  is 
from  B  to  A.  If  we  attend 
only  to  the  direction  of  the 
flow,  and  the  two  vessels  be 
hidden  from  our  view,  then 
obviously  we  can  come  to  no 
conclusion  as  to  the  quantities  FicTT" 

of  water  in  the  two  vessels, 
but  can  only  determine  in  which  the  level  is  higher.  So  with 
two  substances  placed  in  contact :  if  heat  flows  from  A  to  B, 
we  know  that  the  temperature  of  A  is  higher  than  B  ;  the  flow 
of  heat  gives  no  information  as  to  the  respective  quantities  of 
heat  in  A  and  B.  The  flow  of  heat  leads  to  a  more  accurate 
definition  of  temperature. 

The  temperature  of  a  body  is  its  thermal  state  with  refer- 
ence to  its  power  of  communicating  heat  to  other  bodies. 

2.  Expansion. — In  constructing  ordinary  thermometers, 
or  measurers  of  temperature,  advantage  is  taken  of  the  fact 
that  one  effect  of  heat  upon  substances  is  to  change  their 
volume. 

Solids. — Change  in  length  is  allowed  for  in  bridges :  one 
end  at  least  is  not  fixed,  but  is  placed  upon  rollers  to  allow  for 
increase  in  length  as  the  temperature  rises.  For  a  similar 
reason  space  is  left  between  the  ends  of  rails  on  the  railway  \ 
fire-bars  are  fixed  loosely  in  fire-grates  ;  and  we  note  also  that 
telegraph  wires  "  sag ;'  more  in  summer  than  in  winter. 

The  following  experiment  illustrates  linear  expansion. 

An  iron  rod  about  18  inches  long  rests  upon  two  blocks  of 


Heat 


wood  (Fig.  2).  One  end  is  fixed  by  a  heavy  weight,  the  other 
rests  upon  a  fine  needle.  A  straw  is  fixed  at  right  angles  to 
the  needle  with  sealing-wax,  and  behind  the  index  there  is 


FIG.  2. 

a  divided  cardboard  scale.  At  the  beginning  of  the  experi- 
ment arrange  so  that  the  straw  is  vertical,  and  heat  the  bar 
with  a  spirit-lamp ;  the  index  moves  to  the  right,  proving 
that  the  bar  is  expanding,  and  is  rolling  on  the  needle.  The 
iron  bar  can  be  replaced  by  bars  of  brass,  copper,  glass,  etc. 

The  historical  experiment  of  Gravesande  conveniently 
demonstrates  the  cubical  expansion  of  solids. 

A  brass  ball,  a  (Fig.  3),  when  cold,  is  able  to  pass  through 
the  ring  ;//  /  after  heating,  it  is  unable  in  any  position  to  pass 
through. 


FIG.  3. 


PIG. 


Liquids. — The  expansion  of  liquids  can  be  observed  by 
enclosing  them  in  glass  vessels.  A  2-oz.  flask  is  filled  with 
water  coloured  with  red  ink  or  other  colouring  matter  (Fig.  4), 


Heat — Temperature — Thermometers  5 

The  flask  is  closed  with  a  cork,  through  which  passes  a  fine 
tube  about  12  inches  long:  on  inserting  the  cork,  the  water 
rises  slightly  up  the  tube.  A  strip  of  paper  ruled  with  lines  at 
equal  distances,  placed  behind  the  tube,  serves  for  a  scale. 

On  placing  the  flask  in  a  dish  of  warm  water,  you  can 
observe  (a)  a  slight  descent  of  the  liquid,  followed  by  (b)  a 
gradual  ascent. 

The  descent  is  due  to  the  fact  already  ascertained,  that 
solids  expand  on  being  heated.  The  heat  first  affects  the 
flask,  increases  its  volume,  and  there  is  consequently  a  descent 
of  the  liquid.  As  soon  as  the  heat  reaches  the  water,  it  ex- 
pands, and  at  a  greater  rate  than  the  glass  does,  and  there- 
fore the  liquid  rises  up  the  tube.  Liquids,  we  conclude, 
expand  more  than  solids. 

By  fitting  up  two  similar  pieces  of  apparatus,  using  alcohol 
and  mercury  as  the  liquids,  we  can  roughly  compare  the 
expansion  of  water,  alcohol,  and  mercury.  Slightly  adjusting 
the  corks,  we  can  arrange  so  that  at  the  ordinary  temperature 
the  liquids  stand  at  the  same  height  in  each  tube.  Place  all 
three  in  a  basin  of  lukewarm  water.  The  alcohol  rises  higher 
than  the  water,  and  the  water  than  the  mercury ;  observe,  how- 
ever, that  the  mercury  begins  to  rise,  and  reaches  to  its  highest 
point  first. 

In  dealing  with  solids,  high  temperatures  (that  from  flame) 
and  means  of  multiplying  effects  were  necessary  to  demon- 
strate expansion;  in  liquids  the  effect  is  at  once  observable, 
even  when  the  difference  of  temperature  is  not  great  By 
selecting  a  tube  whose  diameter  is  small  compared  with  that 
of  the  flask,  we  can  the  more  readily  observe  the  effects. 
Liquids  enclosed  in  glass  vessels  provided  with  fine  tubes 
are  therefore  used  in  the  construction  of  thermometers. 

3.  Thermometers.  —  The  Mercurial  Thermometer. — A 
clean  tube  (whose  bore  we  shall  assume  is  a  perfect  cylinder) 
is  closed  at  one  end,  and  a  small  bulb,  generally  of  cylindrical 
form,  is  blown.  The  bulb  is  gently  heated,  and  part  of  the  air 
is  expelled ;  the  open  end  is  then  dipped  into  mercury ;  as  the 
air  in  the  tube  cools,  a  small  amount  of  mercury  is  forced  into 
the  bulb.  The  operation  is  repeated  until  sufficient  mercury 


6  Heat 

is  observed  in  the  bulb.  The  bulb  is  then  heated  until  the 
mercury  boils  ;  the  mercury  vapour  drives  out  all  the  air  and 
moisture.  The  end  is  again  dipped  into  mercury  while  heat 
is  being  applied,  and  the  tube  fixed  until  it  cools.  The  pressure 
of  the  atmosphere  forces  the  mercury  into  the  tube,  and  it 
is  found  that  the  bulb  and  tube  are  filled  with  mercury,  and 
mercury  only.  If  any  air  be  present,  the  operation  is  repeated 
until  the  bulb  and  tube  are  completely  filled.  The  glass  is 
carefully  softened  and  drawn  out  where  it  is  intended  to  close 
the  tube ;  when  cold  the  tube  is  cut  off,  leaving  a  very  fine 
aperture. 

We  must  now  determine  the  highest  temperature  the  par- 
ticular thermometer  is  required  to  measure.  For  example,  if 
the  instrument  will  never  be  used  for  any  temperature 
above  boiling  water,  it  may  be  placed  in  a  bath  of  strong 
brine  or  of  acetic  acid  ;  if  intended  to  indicate  higher 
temperatures,  a  bath  of  aniline  may  be  used ;  for  yet 
higher  temperatures,  strong  sulphuric  acid.  The  bath  is 
heated  to  boiling  point,  the  mercury  expands,  and  part 
escapes ;  when  the  bath  boils  steadily  no  more  mercury 
will  be  expelled.  The  fine  end  is  softened  and  closed 
with  a  small  blow-pipe  flame,  the  lamp  is  removed  from 
the  bath,  and  all  is  allowed  to  cool.  The  instrument 
now  contains  mercury  and  mercury  vapour  (Fig.  5).  In 
the  best  thermometers  the  bulb  is  cylindrical. 

4.  The  Fixed  Points.— It  is  found  that  the  tem- 
perature of  melting  ice  is  the  same  whenever  and  wherever 
the  experiment  is  tried  (it  is  affected  by  changes  of 
pressure  (see  §  91),  but  so  slightly  that  the  change  can 
be  neglected  in  constructing  thermometers),  and  that 
the  temperature  of  steam  is  always  constant  if  the 
pressure  be  constant.  These  two  temperatures  determine 
the  two  fixed  points  of  the  thermometer — (a)  the  freezing 
FIG.  5.  pOint  and  (b)  the  boiling  point  of  water. 

(a)  The  Freezing  Point — Clean  ice  is  placed  in  a  vessel 
(preferably  made  of  a  substance  that  is  a  poor  conductor  of 
heat ;  for  example,  of  wood  encased  in  felt)  (Fig.  6).  The 
experiment  must  be  conducted  in  a  room  whose  temperature 


Heat —  Temperature —  Thermometers 


is  above  that  of  melting  ice ;  the  water  formed  escapes  freely  at 

the  bottom  of  the  vessel.     The  closed  thermometer  is  inserted 

BO  that  the  bulb  and  nearly  the  whole 

of  the  stem  is  surrounded  with  ice,  and 

the  whole  is  left  for  about  a  quarter  of 

an  hour.     The  tube  is  then  moved  so 

that  the  mercury  is  just  visible  above 

the  ice.     When   its  position   remains 

steady,  a  scratch  is  made  on  the  glass, 

and  the  point   marked   is   called   the 

freezing  point. 

(b)  The  Boiling  Point.— The  bulb  is 
placed  in  a  metallic  vessel,  M,  with  a 
narrow  upper  part,  A.  This  narrower 
portion  is  surrounded  by  another  vessel, 
B  (Figs.  7  and  8).  The  thermometer 
t  is  passed  through  the  cork  a.  By 
following  the  course  of  the  arrows  in  ~F"IG  6.~ 

the    section,  it    is    seen    that    steam 


FIG.  7. 


^j  j 

FIG.  8. 


8  Heat 

from  the  boiling  water  in  M  surrounds  the  inner  tube  and 
prevents  it  cooling;  the  steam  escapes  by  the  tube  D,  the 
orifice  of  which  should  be  large  enough  to  prevent  any  increase 
of  pressure  inside  the  vessel.  The  small  tube  m,  inserted  in 
B  by  the  cork  E,  contains  mercury,  and  serves  as  a  pressure 
guage :  the  mercury  in  both  limbs  should  be  level  throughout  the 
experiment.  The  temperature  of  boiling  water  depends  upon 
the  substance  of  which  the  vessel  is  made,  and  also  depends 
upon  any  impurities  in  the  water,  but  the  temperature  of  steam 
depends  only  upon  the  pressure;  the  bulb  should  therefore  not 
dip  into  the  water,  but  should  be  slightly  above  it.  The 
pressure  is  measured  by  the  barometer.  After  the  pure  water 
has  boiled  some  time,  the  tube  is  moved  so  that  the  top  of 
the  column  of  mercury  can  just  be  seen;  when  its  position 
remains  steady,  a  mark  is  made  upon  the  glass,  and  the 
barometer  is  read  (the  mercury  in  the  tube  m  should  be  at 
the  same  in  each  branch). 

In  England  the  boiling  point  is  the  temperature  of  steam  at 
the  pressure  of  29*905  inches  of  mercury  at  freezing  point  at  the 
sea-level  in  the  latitude  of  London. 

These  conditions  cannot  ordinarily  be  fulfilled  in  practice, 
and  corrections  must  be  made  (see  Table,  §  99).  The  points 
involved  in  this  definition  of  boiling  point  will  be  discussed 
later;  at  present  we  can  assume  that  the  boiling  point  as 
obtained  in  the  experiment  satisfies  the  conditions. 

5.  The  Scales. — At  the  temperature  of  freezing  point  the 
mercury  fills  the  bulb  and  the  stem  up  to  freezing  point ;  at 
the  temperature  of  the  boiling  point  the  volume  of  mercury 
enclosed  in  glass  expands,  the  increase  in  volume  being  repre- 
sented by  a  thread  of  mercury  extending  from  freezing  point 
to  boiling  point. 

The  distance  between  the  two  fixed  points  is  divided  into 
equal  parts,  called  degrees.  Three  methods  are  followed. 

(i)  Fahrenheit s  Scale. — The  lower  point  is  numbered  32 
(called  32  degrees,  written  32°),  the  upper  212  (written  212°). 
The  distance  between  the  points  is  divided  into  180  equal 
parts,  and  the  graduation  is  continued  above  and  below 
the  points.  The  thirty-second  division  below  the  freezing 


Heat —  Temperature —  Thermometers 


point  is  marked  o,  and  is  called  zero;  below  this  we  have 
minus  i  degree  (written  — 1°),  etc.  It  is  stated  that  Fahren- 
heit took  for  the  zero  of  his  scale  the  lowest  temperature  he 
obtained  by  mixing  ice  and  salt.  This  scale  is  in  common 
use  in  England  and  America  for  ordinary  and  meteorological 
purposes. 

(2)  Centigrade  Scale. — The  boiling  point  is  the  temperature 
of  steam  at  760  millimetres  of  mercury  pressure  at  o°  Centi- 
grade at  the  sea-level  in  latitude  45°  N.,  equivalent  to  the 
29-905  inches  of  mercury  reduced  to  freezing  point  at  the  sea- 
level  in  the  latitude  of  London. 

The  freezing  point  is  called  o°,  the  boiling  point  100°,  and 
the  graduation  is  continued  above  and  below  (—10°,  etc.)  these 
points. 

This  scale  is  used  generally  on  the  continent,  and  in 
England  for  scientific  purposes. 

(3)  Reaumur's  Scale. — Freezing  point  is  marked  o°,  boiling 
point  80°.     This  scale  is  in  use  ^ 

in  Russia  and  in  Germany  for 
commercial  and  household  pur- 
poses. 

Changing  the  readings  from 
one  scale  to  another  is  readily 
done.  Imagine  three  exactly 
similar  thermometers  graduated 
by  the  three  methods,  as  in  Fig.  9. 
In  .order  to  compare  the  readings 
when  the  mercury  is  at  the  height 
x,  y,  we  have,  if  ft  c,  and  r 
represent  the  number  of  degrees 
marked  in  each  scale — 


1 80 


c 

IOO 


r 
80 


210 

/oo 

80 

X 

r           i 

32* 

0* 

0* 

( 

)       « 
r 

)    '( 

G.  9. 

seeing   that   the   numerator  and 

the  denominator  of  each  fraction  represent  the  same  distance. 


io  Heat 

We  easily  deduce  — 

/=£+3» 

5 


r=i(/_32)  r  =  4: 

6.  Calibration  of  the  Tube.—  In  the  construction  of 
the  thermometer  several  assumptions  were  made  that  will  not 
prove  true  in  practice.  In  instruments  of  precision,  therefore, 
corrections  have  to  be  made. 

The  tube  is  not  generally  a  perfect  cylinder;  in  dividing 
the  distance  between  the  fixed  points  equal  lengths  will  not 
necessarily  represent  equal  volumes  ;  to  correct  this  error  the 
tube  is  calibrated.  A  good  tube  should  be  selected,  a  pre- 
liminary trial  being  made  as  follows  :  A  column  of  mercury 
about  2  inches  long  is  drawn  into  the  tube,  and  its  length 
is  exactly  measured  ;  the  column  is  moved  slightly,  and  the 
length  is  again  measured  ;  and  this  process  is  continued  along 
the  tube.  The  mercury  fills  the  same  volume  each  time,  and 
if  there  be  a  variation  in  the  length  exceeding  0*04  inch,  the 
tube  should  be  rejected. 

Into  a  good  tube  a  column  of  mercury  about  £  inch  long 
is  introduced,  say  A  C  (Fig.  io),  the  points  A,  C  are  carefully 


A  C  D 

FIG.  10. 

marked.  The  column  is  now  moved  until  the  end  that  was 
at  A  comes  to  C,  the  other  end  being  at  D.  Evidently  the 
volume  A  C  equals  the  volume  C  D.  The  end  is  then  moved 
to  D,  and  so  on,  until  the  tube  is  divided  into  parts  of  equal 
capacity.  Each  of  these  parts  is  divided  into  a  convenient 
number  of  equal  parts,  it  being  assumed  that  the  diameter 
of  the  tube  remains  uniform  for  any  one  of  the  distances. 
Suppose  such  a  tube  be  made  into  a  thermometer,  and  that 
when  the  two  fixed  points  are  determined,  it  is  found  that  they 
include  between  them  214*5  °f  *ne  tube-divisions,  then  each 


Heat —  Temperature —  Thermometers  1 1 

division  Centigrade  is  equal  to  2*145  °f  tne  tube-divisions, 
Thus  50°  C.  will  be  107*25  from  the  freezing  point,  etc. 

The  following  method  may  be  used  after  the  thermometer 
is  made,  and  is  sufficiently  accurate  for  most  purposes. 

Suppose  o°  and  100°  Centigrade  have  been  determined  in 
the  usual  way.  The  thermometer  is  placed  in  a  temperature 
nearly  50°  C.,  and  the  thread  rises  to  about  C  (Fig.  n).  By 
heating  slightly  at  A,  the  thread  A  C  can  be  separated 
from  the  remaining  mercury  in  the  tube.  Allow  all  to 
cool.  Move  the  thread  so  that  the  end  comes  to  B, 
and  note  the  position  of  the  other  end  at  Q;  move 
the  thread  so  that  one  end  is  at  o  (A),  the  other  being 
at  C2 ;  bisect  the  distance  QCa  in  the  point  C  :  this 
can  be  done  with  sufficient  accuracy  by  the  eye.  Then 
C  is  the  true  50°.  Proceed  similarly  to  divide  the  50°,  Ci[ 
so  as  to  find  25°  and  75°,  and  divide  each  25  into  Cj 
25  equal  parts. 

The  marks  for  the  divisions  are  made  as  follows : 
The  stem  is  covered  with  a  thin  coating  of  wax ;  the 
distance  between  any  two  points  is  divided  by  the  A[ 
dividing-machine  (or  simply  by  placing  the  tube  along 
a  scale) ;  the  machine  removes  the  wax  at  each  point. 
The  whole  is  then  subjected  to  the  vapour  of  hydro- 
fluoric acid,  which  etches  the  glass  where  the  wax  has 
been  removed. 

7.  Variation  of  the  Fixed  Points.— When 
thermometers  are  tested  periodically,  it  is  found  that 
the  zero  changes — its  position  rises  in  the  stem.  The  FlG*  "• 
reason  is  that  glass,  after  being  heated,  does  not  readily  return 
to  its  original  molecular  condition ;  the  greatest  part  of  the  con- 
traction takes  place  as  the  glass  cools,  but  a  small  amount  of 
the  contraction  continues  for  some  time,  and  may  extend 
over  years.  A  thermometer  marked  accurately  at  first,  when 
tested  after  a  few  years  may  show  the  zero  a  few  tenths  of  a 
degree  above  the  original  zero.  If  the  freezing  point  has  risen 
say  0*3°,  then  a  reading  of  20*6°  is  really  20 '3°. 

Again,  if  a  thermometer  be  subjected  to  a  high  tempera- 
ture, say  the  boiling  point  of  water,  and  if  immediately  after 


12  Heat 

the  freezing  point  (or  any  low  temperature)  be  determined, 
the  glass  will  not  have  got  back  to  its  original  state,  and  a 
temperature  noted  by  this  thermometer  will  be  too  low.  It 
is  on  this  account  that  the  freezing  point  is  determined  first. 

These  errors  will  be  diminished  (i)  if  the  glass  be  properly 
annealed;  (2)  if  the  bulbs  be  made  some  time  before  being 
filled ;  and  (3)  if  the  graduation  is  delayed  some  time  after 
the  filling.  The  freezing  point  should,  of  course,  be  determined 
before  the  boiling  point. 

The  thermometer  must  be  used  in  the  vertical  position  if 
the  fixed  points  have  been  determined  while  the  stem  was 
vertical.  Some  thermometers  are  used  in  a  horizontal  posi- 
tion (§  n);  their  fixed  points  must  be  found  in  this  position. 
In  the  vertical  position  the  pressure  of  the  column  of  mercury 
not  only  compresses  the  mercury,  but  also  will  enlarge  the 
bulb,  so  that  the  reading  in  the  horizontal  position  will  always 
be  higher  than  that  in  the  vertical  position. 

8.  Small  Corrections. — In  taking  the  temperature  of  a 
substance,  not  only  the  bulb,  but  also  the  stem  must  be  ex- 
posed to  the  required  temperature,  otherwise  the  reading  will 
be  too  low.    The  method  of  correcting  this  error  will  be  taken 
after  expansion. 

A  thermometer  graduated  at  the  ordinary  pressure,  if 
enclosed  in  a  vacuum,  will  show  a  reading  below  the  true 
reading,  because  relieving  the  pressure  on  the  bulb  causes 
the  bulb  to  expand,  and  thus  the  mercury  sinks. 

In  graduating  the  thermometer  we  assume  that  mercury 
in  glass  expands  the  same  fraction  of  its  volume  per  degree 
when  its  temperature  rises  from  o°  to  100°  C.  as  it  does 
from  10°  to  20°  or  90°  to  100°.  This  is  very  nearly  the  truth 
in  the  case  of  mercury  between  o°  and  1 00°,  and  slightly  above 
and  below  these  temperatures.  In  the  case  of  water  and 
alcohol  this  regularity  does  not  occur  in  their  expansion.  For 
this  reason  a  water  thermometer  graduated  to  agree  with  a 
mercury  thermometer  at  o°  and  80°  C.  would  not  indicate 
40°  C.  when  the  mercury  thermometer  stood  at  40°.  This 
will  be  the  more  readily  understood  after  §§33  and  36. 

9.  Standard    Thermometers,  —  The    graduation    of 


Heat —  Temperature —  Thermometers.  \  3 

thermometers  is  seen  to  be  a  long,  tedious,  and  difficult,  pro- 
cess. Fortunately,  all  the  corrections  are  not  necessary.  The 
work  has  been  carefully  done  on  certain  standard  thermometers 
kept  at  Kew,  and  any  other  thermometer  can  be  compared 
with  a  standard  thermometer  by  subjecting  both  to  the  same 
temperature,  and  marking  on  the  ordinary  thermometer  the 
temperature  indicated  by  the  standard  thermometer,  just  as 
the  alcohol  thermometer  is  compared  with  a  mercury  ther- 
mometer. This  is  analogous  to  testing  ordinary  weights  and 
measures  by  comparing  them  with  the  standard  Government 
weights  and  measures.  If  the  thermometer  be  already  gradu- 
ated, then  a  table  can  be  constructed  showing  how  its  readings 
differ  from  those  of  the  standard  thermometer. 

10.  Alcohol  Thermometer.— Mercury  freezes  at  about 

—  40°  C.,  and  its  rate  of  expansion  for  some  degrees  above  this 
is  not  nearly  equal  to  its  rate  of  expansion  from  o°  to  100°. 
A  mercury  thermometer  cannot,  therefore,  be  used  for  low 
temperatures.     Alcohol  has  not  as  yet  been  frozen ;  it  boils  at 
80°  C.     It  is,  therefore,  a  suitable  liquid  for  low  temperatures. 
The  method  of  construction  is  similar  to  that  of  making  a 
mercury  thermometer.     The  freezing  point  can  be  determined 
as  above.     The  freezing  point  of  mercury,  —40°,  would  give 
another  low  point.     It  is  generally  graduated  by  comparing 
it  with  a  standard  mercury  thermometer  between  60°  C.  and 

—  20°  C,  and  continuing  the  divisions  below  —20°. 


FIG.  12. 

11.  Maximum  and  Minimum  Thermometers.— For 

many  purposes  (mostly  meteorological)  the  highest  and  lowest 


Heat 


FIG.  13. 


temperatures  during  certain  periods  are  needed.     To   effect 
this,  maximum  and  minimum  thermometers  are  used. 

An  ordinary  mercurial  thermometer  is  graduated  when  in  a 
horizontal  position.  It  contains  in  the  tube  a  small  cylinder 
of  iron  (A,  Figs.  12  and  13).  When  the  temperature  rises, 
the  mercury  pushes  the  iron  before  it ; 
when  the  temperature  falls,  as  the 
mercury  does  not  wet  the  iron,  the 
column  retires  and  leaves  the  iron 
behind  it.  The  position  of  the  end 
of  the  iron  nearest  to  the  mercury 
determines  the  maximum  temperature. 

In  the  minimum  thermometer  the  liquid  is  alcohol,  and  the 
index  is  made  of  coloured  glass  (B,  Figs.  12  and  13).  The 
index  is  immersed  in  the  liquid  so  that  the 
concave  surface  of  the  alcohol  touches  one 
end  of  it.  When  the  temperature  rises,  the 
liquid  flows  past  the  index  without  displacing 
it;  when  the  temperature  falls,  the  alcohol 
retreats,  and,  the  concave  surface  not  being 
easily  broken,  pushes  back  the  index. 

The  above  -Rutherford  thermometers  are 
only  useful  when  the  instrument  can  be  kept 
horizontal. 

In  Six's  thermometer  the  tube  is 
bent  as  in  Fig.  14.  The  part  A  is 
filled  with  alcohol,  and  is  cut  off 
from  the  part  B  by  a  thread  of 
mercury.  B  contains  alcohol  and 
alcohol  vapour.  Above  the  mercury 
in  each  branch  of  the  tube  is  a 
small  iron  index  (C  and  D)  covered 
with  glass.  Each  index  slides  easily 
in  the  tube,  but  by  the  aid  of  slight 
springs  of  glass  they  are  kept  upright, 
and  only  move  when  a  slight  pres- 
sure is  exerted  (E,  Fig.  14).  The 
indices  are  brought  into  contact  with  the  columns  of  mercury 


FIG.  14. 


Heat —  Temper  atiire —  Thermometers.  \  5 

by  the  help  of  a  small  magnet.  If  the  alcohol  in  A  expands, 
the  mercury  in  B  rises  and  pushes  forward  the  index.  When 
A  contracts,  the  mercury  falls,  but  the  springs  keep  the  index 
in  position,  the  mercury  easily  flowing  past  it.  Thus  the  lower 
part  of  the  index  D  indicates  a  maximum  temperature,  and 
it  is  readily  seen  that  the  position  of  the  lower  part  of  the  index 
C  indicates  a  minimum  temperature. 

12.  Thermographs. — These  various  forms  lack  the  ac- 
curacy which  is  obtainable  by  the  self-registering  thermometers. 
In  one  form  an  ordinary  thermometer  tube  is  placed  in  front 
of  a  narrow  slit  so  that  light  can  only  pass  where  there  is 
no  mercury.  If  light  from  a  lamp  be  directed  upon  the  slit, 
and  behind  it  there  be  placed  a  cylinder  of  sensitive  paper, 
so  arranged  that  the  paper  is  unwound  at  a  certain  rate,  the 
mechanism  being  regulated  by  clockwork,  then,  when  the 
paper  is  developed,  we  shall  have  the  temperature  indicated 
by  the  thermometer  at  any  given  period.  Thus  in  Fig.  15  we 


FIG.  15. 

see  that  the  maximum  temperature  (point  A)  was  15 '2°  C.  at 
two  o'clock,  the  minimum  (point  B)  was  6°  at  nine  minutes 
past  three.  Maximum  and  minimum  positions  are  also  indi- 
cated at  D  and  C,  and  the  temperature  at  any  particular  time 
is  known.  The  information  is  much  more  definite  than  that 
given  by  the  ordinary  forms. 

For   special  purposes  (deep-sea  soundings,  medical,  etc.) 
special   thermometers    have  been   designed.       Particulars   of 


16 


Heat 


thermometers  (pyrometers)  for  registering  very  high  tempe- 
ratures will  be  given  in  a  later  chapter  (see  Index). 

13.  Differential  Thermometer. — This  thermometer, 
constructed  first  by  Leslie,  measures  differences  of  temperature. 
The  modified  form  shown  in  Fig.  16  is  easily  constructed. 
Two  2-oz.  flasks,  fitted  with  good  indiarubber 
corks,  are  connected  by  a  piece  of  thin  glass 
tubing  24  inches  long,  bent  as  in  Fig.  16. 
A  coloured  liquid  is  drawn  into  the  bend ;  this 
liquid  can  be  adjusted  by  the  aid  of  a  stopper 
in  each  cork  so  that  it  stands  at  the  same 
height  in  each  limb.  The  whole  is  fastened 
to  a  board.  It  can,  if  necessary,  be  adjusted 
by  placing  each  flask  in  a  bath  of  water,  one 
bath  being  5°  higher  in  temperature  than  the 
other.  The  distance  between  the  two  heights 
of  the  liquid  in  each  limb  is  divided  into  five 
equal  parts.  The  instrument  is  used  generally 
to  indicate  differences  of  temperature,  and 
therefore  need  not  be  graduated. 

14.  Mercury  as  a  Thermometric  Sub- 
stance.— Mercury  is  selected  for  thermo- 
meters for  the  following  reasons  : — 

(1)  It  can  be  obtained  pure. 

(2)  Its    expansion    is    almost    regular    for    all    ordinary 
temperatures. 

(3)  It  freezes  at  —40°  C.  and  boils  at  350°  C,  and  is 
therefore  a  liquid  at  all  ordinary  temperatures. 

(4)  It  is  a  good  conductor,  and  therefore  the  whole  of  the 
mercury  soon  attains  the  temperature  of  any  enclosure. 

(5)  It  acts  quickly  (§  2),  and  requires  but  a  small  quantity 
of  heat  to  raise  its  temperature  ;  that  is,  its  specific  heat  is  small. 

15.  Testing  Thermometers.— The  class  thermometers 
should  be  tested.  An  ordinary  tin  with  a  few  holes  punched 
in  the  bottom,  and  filled  with  clean  ice,  will  serve  for  testing 
the  freezing  point.  Afterwards  mix  salt  with  the  ice,  and  test 
again,  and  note  the  effect  of  impurities. 

For  verifying  the  boiling  point,  a  flask  with  a  long  neck 


FIG.  16. 


Heat —  Temperature —  Thermometers.  \  7 

and  half  filled  with  clean  water  may  be  used  (Fig.  17).     Pass 
a  thermometer  through  a  cork,  through  which  also  passes  a 
bent  open  tube ;  the  thermometer  must  not  touch  the  water ; 
it    should   be   merely    sur- 
rounded with  steam.  (What 
are  the  defects  of  this  ar- 
rangement? Compare  Figs. 
7  and  8.) 

Add  salt  to  the  water, 
and  again  notice  the  height 
of  the  mercury  when  the 
thermometer  is  immersed 
in  the  steam.  Place  the 
thermometer  in  the  salt  and 
water,  and  again  observe 
the  temperature.  Try  mix- 
tures of  calcium  chloride 
and  water. 

To  observe  the  in- 
fluence of  pressure,  con- 
nect the  tubes  of  A  and  B. 
The  tube  B  dips  into  water, 
and  therefore  the  pressure 
inside  the  flask  is  greater  than  that  of  the  atmosphere.  Re- 
place the  water  in  C  with  mercury,  and  test  again. 


FIG.  17. 


EXAMPLES.    I. 

1.  Define  "heat"  and  "temperature." 

2.  Describe  the  construction  of  a  mercurial  thermometer.    How  are  the 
fixed  points  determined? 

3.  What  are  the  advantages  and  disadvantages  of  mercury  over  water 
as  a  thermometric  substance,  say  between  10°  C.  and  40°  C.  ? 

4.  Change  the  following  readings  on  the  Centigrade  scale  into  readings 
on  a  Fahrenheit  and  Reaumur  scale:  100°,  o°,  -273°,  —38°,  200°. 

5.  The  following  readings  are  from  a  Fahrenheit  thermometer :  what 
will  they  be  on  a  Centigrade  thermometer:  212°,  32°,  —32°,  —460°? 

6.  Why  is  steam  used  rather  than  water  in  determining  the  upper  fixed 
point  ? 

7.  A  thermometer  with  a  uniform  tube  has  12  F.  divisions  in  an  inch  ; 
how  many  C.  divisions  would  there  be  in  an  inch  ? 


1 8  Heat 

8.  You  test  a  cheap  thermometer  (C. ),  and  find  that  the  true  freezing 
point  is  +1°,  and  the  true  boiling  point  is  101° :  what  is  the  real  value 
of  50°  on  such  a  thermometer  ? 

9.  A  thermometer  whose  bulb  alone  is  immersed  in  a  liquid  reads  44°  F. 
Is  this  too  high  or  too  low — give  reasons  for  your  answer — in  two  cases, 
(a)  when  the  temperature  of  the  air  is  60°  F.  ;  (b)  when  it  is  32°  F.  ? 

10.  A  thermometer  tube  is  given  to  you  closed  in  the  usual  way,  but 
without  any  markings  :  how  would  you  determine  the  fixed  points,  and 
what  plan  would  you  follow  in  calibrating  the  tube  ? 


CHAPTER   II. 
EXPANSION  OF  SOLIDS. 

16.  Coefficient  of  Linear  Expansion. — The  general 
effect  that  solids  as  a  rule  expand  when  their  temperature  is 
raised,  and  contract  when  their  temperature  falls,  has  already 
been  referred  to. 

When  we  consider  the  change  in  length,  as  in  estimating 
the  expansion  of  a  rod  or  of  a  telegraph  line,  we  deal  with 
linear  expansion  ;  if  change  of  area  is  noted,  we  deal  with 
areal  or  square  expansion ;  and  in  change  of  volume  we  deal 
with  cubical  expansion. 

If  a  body  be  perfectly  isotropic,  the  change  in  length  in 
any  direction  will  be  the  same ;  in  crystalline  bodies  the 
change  in  different  directions  will,  in  general,  be  different. 

The  coefficient  of  linear  expansion  for  i°  is  the  ratio  of 
the  increase  in  length  when  the  temperature  is  raised  i°  to  the 
original  length. 

Thus  if  a  bar  of  iron  be  4  feet  long  at  10°  C.,  and  4  feet 
0*0472  inch  at  100°  C.,  the  increase  in  length  is  0-00393  foot 
for  90°,  and  0*0000433  f°ot  f°r  l0-  Therefore  its  coefficient 
of  expansion  for  i°  is  0-0000433  -4-  4,  or  0*0000  n  nearly. 

This  definition  would  be  sufficient  if  the  coefficient  were 
the  same  for  a  given  substance  when  measured  at  any  tempera- 
ture. Measurable  differences,  however,  occur,  so  that  the  more 
correct  definition  is — 

The  coefficient  of  linear  expansion  at  any  given  tempera- 
ture is  the  ratio  of  the  increase  of  length  when  the  temperature 
is  raised  one  degree  to  the  original  length  at  the  given  tem- 
perature.. 


2O 


Heat 


In  the  Centigrade  scale  the  coefficient  is  frequently  defined 
so  that  the  given  temperature  is  o° ;  in  the  Fahrenheit  scale 
it  is  frequently  defined  at  60°. 

Ordinarily,  the  coefficient  is  deduced  from  experiments 
where  the  range  of  temperature  is  100°  or  more,  and  the 
coefficient  is  determined  by  dividing  the  ratio  by  100  or  the 
number  of  degrees ;  that  is,  it  is  assumed  that  the  coefficient  of 
expansion  is  the  same  at  all  temperatures. 

17.  Lavoisier  and  Laplace's  Method. — A  brass  trough 
rests  upon  a  surface  between  four  stone  supports  (Figs.  18  and 


FIG.  18. 

19).  On  two  of  the  supports  is  an  axis,  carrying  firmly  a 
telescope,  G ;  a  glass  rod,  D,  is'  securely  fixed  to  the  middle 
of  the  axis,  at  right  angles  to  it.  Both  rod  and  telescope 
move  with  the  axis.  The  other  two  supports  also  carry  a 
bar,  to  which  a  glass  rod,  F,  is  fixed  at  right  angles.  This  rod 
is  kept  firmly  in  its  place  by  a  stop. 

The  bar  K  H,  whose  expansion  is  to  be  measured,  is  placed 
on  glass  rollers  in  the  trough,  one  end  being  firmly  fixed  against 


•ff 


FIG.  19. 


the  glass  rod  F.    The  trough  is  first  filled  with  melting  ice,  and 
the  telescope,  which  is  fitted  with  the  ordinary  cross-wires,  is 


Expansion  of  Solids  21 

directed  towards  a  distant  scale  and  the  reading  A  is  taken, 
care  being  taken  that  the  glass  rod  D  presses  against  the  end 
H.  The  ice  is  removed,  and  the  trough  filled  with  oil  or 
water,  which  is  heated.  The  bar  expands.  The  end  K  being 
fixed,  the  movement  is  registered  by  the  end  H,  which  moves 
to  C,  moves  the  rod  D,  and  thus  depresses  the  telescope,  and 
the  distant  scale  is  again  read  at  B. 

At  the  beginning  of  the  experiment  it  was  arranged  so  that 
the  angles  BAG  and  GHC  were  right  angles.  The  angle 
AGB  =  angle  HGC.  Therefore  the  triangles  AGB  and  HGC 
are  similar. 

HC_GH 
*'•  AB  ~~AG 

GH  and  AG  were  carefully  measured  at  the  beginning,  and 

C*  T-T  T 

were  constant  for  each  experiment.    In  a  series  -  T  =  - 

AG       744 


. 

744 

AB  is  read  off  on  the  scale. 

On  dividing  HC  by  the  original  length  of  the  bar,  and  by 
the  number  of  degrees  through  which  the  bath  is  heated,  we 
obtain  the  coefficient  of  expansion. 

The  difficulty  consists  in  determining  exactly  where  the 
bar  touches  G  H,  and  measuring  from  this  point  to  the  optical 
axis  of  the  telescope.  More  satisfactory  results  are  obtained 
when  the  elongation  is  measured  by  means  of  a  micrometer 
screw.  If  a  screw  is  made  with  say  a  hundred  turns  to  the 
inch,  and  works  in  a  fixed  socket,  then  each  turn  of  the  head 
indicates  that  the  screw  has  moved  forward  a  distance  of  o'oi 
inch  ;  if  the  circumference  of  the  head  of  the  screw  be  divided 
into  a  hundred  parts,  then,  if  the  head  be  turned  through  one 
division,  a  distance  is  indicated  of  o'oooi  inch.  The  accuracy 
of  the  instrument  depends  upon  its  construction. 

18.  Roy  and  Ramsden's  Method.—  The  following  is  a 
modification  of  Roy  and  Ramsden's  method  :  Two  bars  01 
equal  lengths  of  cast  iron  are  placed  in  two  troughs,  A  and  C 
(Fig.  20).  The  bars,  each  about  6  feet  long,  are  surrounded 


22 


Heat 


with  melting  ice.  Vertical  rods  are  fixed  at  the  ends  of  these 
bars ;  on  the  rods  in  one  bar  are  placed  two  telescopes  with  the 
usual  cross-wires ;  on  those  of  the  other  bar  are  two  cross-wires 
suitably  illuminated. 

The  centres  of  the  cross-wires  and  the  optical  axes  of  the 
eye-pieces  are  a  known  equal  distance  apart,  a  distance  that 


77, 


FIG. 20. 

remains  constant  during  the  experiment,  seeing  that  the  bars 
remain  at  freezing  point.  Between  these  troughs  is  a  third 
trough,  B,  containing  the  bar  whose  elongation  is  to  be  measured. 
It  carries  similar  vertical  rods,  each  carrying  cross-wires,  n  and 
m.  The  cross-wire  n  is  movable,  by  means  of  a  micrometer 
screw,  along  the  axis  of  its  bar.  The  middle  bar  is  surrounded 
with  melting  ice,  and  the  apparatus  is  arranged  so  that,  on 
looking  through  the  telescopes  of  C,  the  cross-wires  in  A  and  B 
are  clearly  seen  one  behind  the  other.  Under  these  conditions 
the  distance  between  the  intersections  of  the  cross-wires  in  B 
is  the  known  distance  between  the  centres  of  the  cross-wires 
in  A  or  the  centres  of  the  cross-wires  in  the  telescopes.  The 
trough  B  is  now  heated  to  the  temperature  of  boiling  water 


Expansion  of  Solids  23 

or  boiling  oil,  the  exact  temperature  being  indicated  by  the 
thermometers  /,  /.  The  end  of  the  rod  near  m  is  kept  fixed,  so 
that  all  the  elongation  is  effected  at  n.  The  agreement  between 
the  cross-wires  is  now  disturbed,  the  micrometer  screw  a  is 
turned,  and  the  turns  and  parts  of  a  turn  noted  necessary  to 
bring  the  cross-wire  n  into  agreement  again.  This  measures 
the  elongation,  and  the  coefficient  of  expansion  can  be  cal- 
culated. 

19.  Direct  Method. — The  following  method  is  described 
in  Glazebrooke  and  Shaw's  "  Practical  Physics."  Its  advantage 
is  that  the  measurements  are  taken  direct 

The  reading  microscopes  described  stand  vertically,  and,  by 
turning  a  micrometer  screw,  move  horizontally.  The  movement 
is  registered  by  an  index  moving  along  a  scale  divided  into 
millimetres,  the  distance  between  two  threads  of  the  screw  is 
also  i  mm.,  so  that  each  complete  turn  of  the  screw  moves  the 
microscope  one  millimetre.  The  head  of  the  screw  is  divided 
into  100  parts,  therefore  moving  it  through  one  part  moves  the 

microscope,   and  therefore  the  index,  through  i  mm.  X  — 

=  o'oi  mm.  The  apparatus  is  so  constructed  that  when  the 
zero  of  the  screw  is  at  the  mark,  the  index  is  on  one  of  the 
divisions. 

If  the  index  on  one  microscope  be  between  the  forty-fifth 
and  forty-sixth  divisions,  and  the  screw-head  denotes  76,  then1 
the  position  denoted  is  4576  mm.,  or  4*576  cm. 

"  We  require  to  measure  the  length  of  a  rod,  or  the  dis- 
tance between  two  marks  on  it,  at  two  known  temperatures, 
say  15°  C.  and  100°  C. 

"The  highest  degree  of  accuracy  requires  complicated 
apparatus.  The  following  method  is  simple,  and  will  give 
very  fair  results  : — 

"A  thick  straight  rod  is  taken,  about  50  cm.  in  length,  and 
a  glass  tube  of  4  or  5  cm.  bore  and  somewhat  greater  length 
than  the  rod.  The  tube  is  closed  with  a  cork  at  each  end, 
and  through  each  cork  a  small  piece  of  glass  tubing  is  passed, 
and  also  a  thermometer.  Two  fine  scratches  are  made  on  the 
rod,  one  close  to  each  end,  at  right  angles  to  its  length,  and 


24  Heat 

two  other  scratches,  one  across  each  of  the  former,  parallel  to 
the  length.  The  glass  tube  is  clamped  in  a  horizontal  position 
and  the  rod  placed  inside  it,  resting  on  two  pieces  of  cork  or 
wood  in  such  a  manner  that  the  scratches  are  on  the  upper 
surface  and  can  be  seen  through  the  glass.  The  whole  should 
rest  on  a  large  stone  slab  — a  stone  window-sill  serves  admirably. 

<f  The  piece  of  glass  tubing  in  one  of  the  corks  is  con- 
nected with  a  boiler  from  which  steam  can  be  passed  into  the 
tube;  the  other  communicates  with  an  arrangement  for  con- 
densing the  waste  steam. 

"  A  pair  of  reading  microscopes  are  "then  brought  to  view 
the  cross-marks  on  the  rod,  and  are  clamped  securely  to  the 
stone.  The  microscopes  should  be  placed  so  that  they  slide 
parallel  to  the  length  of  the  rod ;  this  can  be  done  by  eye  with 
sufficient  accuracy  for  the  purpose. 

"  For  convenience  of  focussing  on  the  rod  which  is  in  the 
glass  tube,  the  microscopes  must  not  be  of  too  high  a  power. 
Their  supports  should  be  clamped  down  to  the  stone  at  points 
directly  behind  or  in  front  of  the  position  of  the  microscopes 
themselves,  to  avoid  the  error  due  to  the  expansion  of  the 
metal  slides  of  the  microscopes,  owing  to  change  of  temperature 
during  the  experiment. 

"Call  the  microscopes  A  and  B;  let  A  be  the  left-hand 
one  of  the  two,  and  suppose  the  scale  reads  from  left  to  right. 
Turn  each  microscope  tube  round  its  axis  until  one  of  the 
cross-wires  in  the  eye-piece  is  at  right  angles  to  the  length  of 
the  rod,  and  set  the  microscope  by  means  of  the  screw  until 
this  cross-wire  passes  through  the  centre  of  the  cross  on 
the  rod. 

"Read  the  temperature,  and  the  scale  and  screw-head 
of  each  microscope,  repeating  several  times.  Let  the  mean 
result  of  the  readings  be — 

Temperature.  A  B 

15°  C.         ...         5-106  cm.         ...         4738cm. 

"  Now  allow  the  steam  to  pass  through  for  some  time ; 
the  marks  on  the  copper  rod  will  appear  to  move  under 
the  microscopes,  and  after  a  time  will  come  to  rest  again. 


Expansion  of  Solids  25 

"  Follow  them  with  the  cross-wires  of  the  microscopes,  and 
read  again.  Let  the  mean  of  the  readings  be— 

Temperature.  A  B 

100°  C.         ...         5-074  cm.         ...         4780  cm. 

"  Then  the  length  of  the  rod  has  apparently  increased  by 
5-106  -  5-074  +  478  -  4738>  <>r  °'°74  cm. 

"The  steam  will  condense  on  the  glass  of  the  tube  which 
surrounds  the  rod,  and  a  drop  may  form  just  over  the  cross 
and  hide  it  from  view.  If  this  be  the  case,  heat  from  a  small 
spirit-flame  or  Bunsen  burner  must  be  applied  to  the  glass  in 
the  neighbourhood  of  the  drop,  thus  raising  the  temperature 
locally  and  causing  evaporation  there. 

"  Of  course,  the  heating  of  the  rod  and  tube  produces  some 
alteration  in  the  temperature  of  the  stone  slab,  and  causes  it  to 
expand  slightly,  thus  producing  error.  This  will  be  very  slight, 
and  for  our  purpose  negligible,  for  the  rise  of  temperature  will 
be  small  and  the  coefficient  of  expansion  of  the  stone  is  also 
small. 

"  We  have  thus  obtained  the  increase  of  length  of  the  rod 
due  to  the  rise  of  temperature  of  85°.  We  require  also  its 
original  length. 

"  To  find  this  remove  the  rod  and  tube  and  replace  them 
by  a  scale  of  centimetres,  bringing  it  into  focus.  Bring  the 
cross- wires  over  two  divisions  of  the  scale,  say  10  and  60, 
and  let  the  readings  be — 

A  B 

4-576  cm.  5-213  cm. 

Then  clearly  the  length  of  the  rod  at  15°  C.  is- 

50  -  (5-106  -  4'576)  +  (4738  -  5<213)>  or  48'995  cm- 
"To  find  the  coefficient  of  expansion,  we  require  to  know 
the  length  at  o°  C.     This  will  differ  so  little  from  the  above 
that  we  may  use  either  with  all  the  accuracy  we  need,  and  the 

required  coefficient  is  5—  — ,  or  0*0000178. 

55  x  40*995 

"  Experiment. — Determine  the  coefficient  of  expansion  of 
the  given  rod. 


26  Heat 

11  Enter  results  thus — 

Increase  of  length  of  rod  between  15°  and  100°  C.     0*074  cm. 

Length  at  15°      ...         ...         ...         ...         ...  48*995  cm. 

Coefficient  ...         ...         ...         ...         ...      0*0000178" 

(See  Worked  Examples,  p.  32,  No.  2.) 

20.  Exceptions. — Generally  bodies  expand  when  heated. 
Stretched  indiarubber,  iodide  of  silver,  and  a  few  other  sub- 
stances contract  when  their  temperature  is  raised.     Their  co- 
efficient of  expansion  is  negative.     If  a  tube  of  indiarubber 
be  slightly  stretched,  and  steam  be  passed  through,  the  con- 
traction  can    be   readily    observed;    if   the   indiarubber   be 
suddenly  stretched,  its  temperature  rises. 

21.  Results. — The  coefficients  of   linear  expansion   are 
small  quantities,  and  can  only  apply  to  the  particular  specimens 
experimented  upon.     The  coefficient  will  vary,  for  example, 
with  particular  specimens  of  iron,  and  the  numbers,  as  will 
have  been  seen  by  the  experiments,  are  generally  calculated 
by  measuring  the  expansion  for  100°,  and  thus  measure  the 
average  expansion  between  two  given  temperatures. 

TABLE  OF  COEFFICIENTS  OF  LINEAR  EXPANSION  FOR  i°  C. 

Glass          ...  =  0-0000085  Copper  ...  =  0*000017 

Platinum    ...  =  00000085  Brass  ...  =  0*000019 

Steel  ...  =  0*000012  Silver  ...  =  0*000019 

Cast  iron    ...   =  o'oooon  Tin  ...  =  0*000022 

Wrought  iron  =  0*000012  Lead  ...  =  0*000029 

Gold          ...  =  0*000015  Zinc  ...  =  0*000029 

The  values  for  the  Fahrenheit  scale  will  be  f  of  these  for 
the  Centigrade  scale. 

22.  Practical  Applications.— The  change  in  length  due 
to  change  of  temperature  in  solid  bodies  is  exceedingly  small, 
and   in   practice   may   generally   be    neglected.       For    exact 
purposes,  such  as  determining  the  standard  measures,  it  must 
be  observed. 

The  standard  yard  is  defined  as  the  distance  between  the 
centres  of  the  transverse  lines  in  the  two  gold  plugs  in  the 
bronze  bar  deposited  in  the  office  of  the  Exchequer  at  62°  F. 


Expansion  of  Solids  27 

In  the  case  of  the  French  metre,  the  length  is  to  be  taken  at 
the  temperature  of  melting  ice. 

In  bridges,  as  has  already  been  observed,  the  expansion 
must  be  allowed  for;  not  only  do  the  substances  expand 
or  contract,  but  they  do  work  in  so  doing.  An  iron  rod,  for 
example,  i  square  inch  in  section,  in  cooling  through  9°  C, 
would  exert  a  force  of  i  ton.  The  force  of  contraction  is 
illustrated  in  Fig.  21.  An  iron  bar,  A  B,  passes  through 


33 

g 

D 

? 

1        .-                            _                   ! 

FIG.  21. 


sockets  in  a  strong  cast-iron  frame,  C  D.  The  iron  bar  has 
a  hole  in  one  end,  B,  through  which  passes  a  cast-iron  rod,  F  ; 
at  the  other  end  is  a  screw-thread,  on  which  a  nut,  N,  with 
two  arms,  works.  The  rod  is  heated,  placed  in  the  sockets ; 
the  rod  F  is  inserted,  and  the  nut  screwed  up  tightly.  As  the 
temperature  falls,  the  bar  contracts,  and  the  force  is  sufficient 
to  break  the  small  bar  F. 

By  passing  iron  bars  through  buildings,  fixing  screws  to 
the  ends,  and  tightening  these  screws  when  the  bars  are  heated, 
it  has  been  possible  to  straighten  walls  that  have  bulged  out ; 
the  force  exerted  on  cooling  being  sufficient  to  gradually  bring 
the  walls  to  their  original  position. 

A  striking  illustration  is  afforded  in  the  case  of  telegraph 
wires.  An  iron  wire  stretched  across  a  span  of  400  feet,  with 
a  "sag"  of  5  feet  at  the  temperature  25°  C,  would  have 
a  strain  upon  it  of  13,544  Ibs.  per  square  inch.  In  winter, 
at  a  temperature  of  —  5°  C.,  the  strain  would  be  34,000  Ibs.  per 
square  inch,  sufficient  to  permanently  stretch  the  wire. 

The  effect  of  change  of  temperature  on  the  strength  of 
materials  is  important  in  practical  work,  especially  in  the  con- 
struction of  boilers.  Cast  iron,  for  example,  loses  strength 
when  heated  above  120°  F. ;  wrought  iron  seems  constant  up 
to  400°  F.,  it  then  declines,  until  at  red  heat  its  strength 
has  fallen  from  20  tons  to  15^  tons  per  square  inch. 


28 


Heat 


23.  Compensating  Pendulums. — The  number  of  swings 
made  by  the  pendulum  of  a  clock  in  a  given  time  varies 
inversely  as  the  square  root  of  the  length  of  the  pendulum. 
It  is  therefore  necessary  to  devise  methods  to  keep  the  length 

of  the  pendulum  constant  during 
changes  of  temperature.  In  Harri- 
son's gridiron  pendulum  (Fig.  22) 
the  bob  is  supported  by  framework, 
as  in  the  figure.  The  five  shaded 
portions  (d,  ey  /,...,  and  the  piece 
b)  are  steel;  the  others  (c,  n,  .  .  .) 
are  brass.  An  examination  of  the 
figure  shows  that  the  steel  (fixed 
at  the  top),  expanding,  will  lower 
the  bob,  while  the  expansion  of 
the  brass  (fixed  at  the  bottom)  will 
lift  the  bob. 

From  the  Table,  p.  26,  the  co- 
efficients of  expansion  of  steel  and 
brass  are  as  12  :  19.  Then  evi- 
dently the  lengths  of  steel  and  brass 
must  be  inversely  as  these  num- 

.       .     b  +  i  +  e  -\-  d       19 
bers ;  that  is,  -  -  =  — . 

n  +  c  12 

So  constructed,  the  distance  of  the 
centre  of  the  "bob"  from  the 
point  of  suspension  will  be  constant 
at  all  temperatures.  It  is,  per- 
haps, needless  to  say  that  most  of 
the  compensating  pendulums  in 
ordinary  clocks  are  mere  toys. 

If  strips  of  brass  and  zinc  be  soldered  together,  and  be 
then  heated,  the  strip  bends.  The  coefficients  of  expansion 
of  brass  and  zinc  are  as  19  :  29;  therefore,  in  expanding,  the 
zinc  forms  the  convex  side  of  the  compound  bar. 

Advantage  is  taken  of  this  in  constructing  the  balance- 
wheels  of  chronometers. 

The  rate  of  a  chronometer  depends  upon  the  mass  of  the 


FIG.  22. 


Expansion  of  Solids  29 

balance-wheel  and  the  distance  of  the  circumference  from  the 
centre.  The  parts  B,  C  (Fig.  23)  are  made  up  of  a  compound 
strip,  the  metal  with  the  highest  coefficient 
being  on  the  outside.  When  the  tempera- 
ture rises,  the  radius  A  expands,  and  the 
chronometer  would  lose  time,  but  the  heat 
causes  the  strips  B,  C  to  curve  inwards. 
The  masses  D  are  thus  brought  nearer 
the  centre,  and  this  compensates  for  the 
expansion  of  A. 

24.  Coefficient  of  Square  Expansion.— The  coeffi- 
cient of  square  expansion  is  the  ratio  of  the  increase  in  area 
for  one  degree  to  the  original  area. 

If  a  square  of  unit  side  expands  for  one  degree  so  that  the 
side  becomes  (i  +  k),  k  being  numerically  equal  to  the  co- 
efficient of  linear  expansion,  the  area,  if  the  body  be  isotropic, 
will  be  (i  +  k)~,  and  the  coefficient  of  expansion  for  square 

2k  -f-  k~ 

expansion  will  be  -  — .  But  k  being  small,  &  can  be  neg- 
lected, and  we  have  the  coefficient  of  areal  expansion  =  2k; 
i.e.  twice  the  coefficient  of  linear  expansion.  If  the  coefficient 
of  expansion  in  one  direction  be  kly  in  the  other  k^  then 
the  new  area  =  (i  +  *i)  (i  +  *»)  =  i  +  *i  +  ^  +  *A- 

kfa  can    be  neglected,   and    the   coefficient   equals  - 

=  k^  -f-  &,  the  sum  of  the  coefficients  of  linear  expansion, 

An  examination  of  the  Table,  p.  26,  shows  that  the  linear 
expansions,  and  therefore  the  areal  and  cubical  expansions,  of 
glass  and  platinum  are  equal.  It  is  this  fact  that  makes  it 
possible  for  platinum  wire  to  be  fused  into  glass.  If  the 
expansion  (areal)  were  greater,  then,  on  cooling,  either  the 
glass  would  crack  or  the  tube  would  not  be  air-tight 

25.  The  Coefficient  of  Cubical  Expansion  is  the 
ratio  of  the  increase  in  volume  for  one  degree  to  the  original 
volume. 

If  the  body  be  isotropic,  then  a  cube  whose  side  is  unity 
will  have  a  side  i  -f  k  when  the  temperature  rises  i°.  There- 
fore the  new  volume  =  i  -f  $k  -f  3^  +  1?.  &  and  similarly  & 


Heat 


FIG.  24. 


are  very  small  compared  with  k,  and  may  be  neglected,  so  that 
the  coefficient  of  cubical  expansion  =  3    =  $k ;  i.e.  three  times 

the  coefficient  of  linear  expansion.    Imagine  the  cube  with  thick 

lines  to  be  i  foot  each  side  (Fig.  24). 
After  expansion  the  complete  cube  is 
formed,  made  up  as  regards  volume, 
of  the  original  cube  +  3  square 
slabs  (AJ,  Ao,  A3  are  the  diagonals ; 
volume  of  each  is  i  x  r  X  d)  -f  3 
strips  (a  section  of  each  is  shown ; 
volume  of  each  is  i  X  d  x  d)  -f  a 
sma^  cu^e  whose  volume  is  d  x  d 
X  d.  This  illustrates  that  (i  +  Kf 
=  i  +  3*  +  3*3  +  k\  We  are 
able  to  neglect  1?  and  k?,  because 
they  are  so  small  compared  with  k.  For  solid  bodies  k  is  less 
than  0*00003,  so  that  for  a  change  of  100°  we  should  have  3^ 

less  than  o '0000002 7,  or    700QQO  of  the  total. 

If,  therefore,  the  solid  be  isotropic,  or,  being  crystalline, 
if  its  coefficients  of  linear  expansion  be  the  same  in  all 
directions,  the  coefficient  of  cubical  expansion  can  be  calcu- 
lated from  the  coefficient  of  linear  expansion. 

In  many  crystals  the  coefficients  of  expansion  in  the 
direction  of  three  axes  are  not  equal ;  the  result  of  increase  of 
temperature  is  therefore  distortion  of  shape.  The  coefficient 
of  cubical  expansion  is  in  these  cases  the  sum  of  the  three 
coefficients. 

26.  Small  Quantities. — The  coefficient  of  expansion 
numerically  equal  to  the  expansion  of  unit  length  when  the 
temperature  is  raised  one  degree  is  small  compared  with  unity ', 
and  therefore  the  second  and  third  powers  can  generally  be 
neglected.  This  has  already  been  shown  to  be  the  case  for 
solids.  The  same  powers  can  generally  be  neglected  in  liquids. 
In  gases  we  shall  see  the  coefficient  is  generally  about  the 
value  0*003665  (compare  this  with  0*00003);  the  degree  of 
accuracy  required  must  determine,  in  questions  relating  to 
gases,  the  powers  that  may  be  disregarded. 


Expansion  of  Solids  31 

Generally  let  dt  e,  and /be  quantities  small  compared  with 
unity. 


Approximation  to  the 

First  order 
of  small 
|     quantities. 

Second  order 
of  small 
quantities. 

(l-d)-  =  l—2d+d*                              I-2.-/ 

I  -^-2d-\-d 
I  —  2d-\-d* 

(i+a)(i+e)  = 


(I  -</)»=  i  -y+  #»-< 


L  =  i  +</_rt-2-r-</8+etc. 


I—  a 


i-d 


i+d 


i+<t+e+f+<& 
i-d+d* 
i+d-d* 
^d-e-de+t* 


27.  Density  and  Temperature. — If  LO  be  the  length 
of  a  bar  at  o0°,  and  k  be  the  coefficient  of  linear  expansion, 
then  at  r*  the  length  is  LO(I  +  £/),  and 

L,  -  L0(i  +  kt) 
If  V  represent  volume — 

V,  =  V0(i  +  3^) 

The  density  of  any  substance  at  any  temperature  is  the 
mass  of  unit  volume  at  that  temperature. 

If  Vo,  Vw  VT  be  the  volumes  of  a  substance  at  o°,  /°,  and 
T°,  and  DO,  Dw  DT  be  the  densities  of  the  substance  at  o°,  /°, 
and  T°,  the  mass  of  the  substance  is  the  product  of  volume 
and  density,  and,  of  course,  does  not  change  with  expansion. 

/.  mass  =  M  =  V0D0  =  V,D,  =  VTDT 
That  is— 

Vt :  VT : :  DT  :  D, 

or  the  densities  are  inversely  as  the  volumes.     If  K  =  coeffi- 
cient of  cubical  expansion — 


32  Heat 


Vt  =  V0(i  +  K/) 
.-.  DT  :  D,  -  i  +  K/  :  i  +  KT 


if  K  be  small  compared  with  unity.    In  the  special  case  where 
t  =  o— 

DT  =  D0  i  +*KT  -  D0(i  -  KT) 
if  K  be  small  compared  with  unity. 

WORKED  EXAMPLES. 

1.  A  rod  of  copper  measures  10  feet  at  o°  C.  ;  its  length  at  100°  C.  is 
10  feet  0*191  inch  :  find  the  coefficient  of  expansion  of  copper. 

The  increase  in  length  for  100°  =  0*191  inch  =  0*0159  foot 

„  i°  =0-0001  59  foot 

increase  in  length  for  i°       0-000159  ft. 
.-.  coefficient  =  --  original  length  ~^^W.  -  =  0-0000159 

=  o  'oooo  1  6  nearly 

2.  See  example  at  the  end  of  §  19  . 

The  result  obtained  is  the  mean  coefficient  between  15°  and  100°  (the 
correct  integers  are  17769).  We  might  consider  it  more  accurately  thus: 
If  k  be  the  coefficient  of  linear  expansion  — 

L15  =  L0(i  +  15*)  L100  =  L0(i  +  ioo*) 

.   L100  _  49*069  _  i  +  IPO*  A. 

'  L15       48-995        1  +  15* 

.-.  49-069  +  15  x  49-069*  =  48-995  +  4899*5^ 
/.  *(4899'5  -'736-035)  =0-074 


Now,  obviously  the  errors  of  the   experiment  are  greater  than   any 
meaning  we  can  attach  to  figures  in  the  seventh  and  eighth  places  ;  there 

is  thus  no  advantage  in  the  latter  method.    If  at  step  A  we  write  — 


i.e.  the  result  by  the  approximate  method. 

3.  A  rod  is  measured  at  75°  F.  ;  by  the  brass  scale  used  its  length  is 
18-04  inches;  the  scale  was,  however,  graduated  at  the  temperature  of 
melting  ice,  and  its  mean  coefficient  of  expansion  between  melting  point 


Expansion  of  Solids  33 

and  freezing  point   is  o  '0000105   per  degree   Fahrenheit  :   find  the  real 
length  of  the  rod. 

The  brass  scale  has  expanded,  and  therefore  indicates  too  low  a 
measurement. 

i"  on  the  scale  at  32°  F.  measures  (i  +  0-0000105  X  43)"  =  1-0004515" 
at  75°  F. 

/.  the  true  length  of  i"  at  75°  F.  is  ^^         inch 

.*.  the  true  length  of  the  rod  =  18*04  —  —  -  -  =  18*032  inches 

i  '00045  '  5 
practically  a  negligible  quantity. 

4.  The  volume  of  a  leaden  ball  at  60°  F.  is  100  cubic  inches  :  find  its 
volume  at  the  boiling  point  of  water  (mean  coefficient  of  linear  expansion 
on  the  Fahrenheit  scale,  0-0000157). 

Coefficient  of  cubical  expansion  (K)  =  O'OOOO47I 

.*.  V2i2  =  V60(i  +  152  X  0-0000471) 
=  100(1-0071592) 
=  100*716  cubic  inches 

EXAMPLES.    II. 

1.  Three  rods  of  glass,  copper,  and  silver,  are  each  lofeet  long  at  O°C.  : 
find  their  lengths  at  100°  C. 

2.  The  coefficient  of  expansion  for  a  specimen  of  English  flint  glass 
was  found  to  be  0-0000045  1  Per  degree  Fahrenheit  :   compare  the  density 
of  such  glass  at  32°  F.  and  at  212°  F. 

3.  The  diameter  of  a  tin  sphere  at  15°  C.  is  10  cm.:  find  its  volume 
at  85°  C. 

4.  One  end  of  a  cast-iron  girder  in  a  factory  is  fixed  :  what  play  must 
be  allowed  the  other  end  when  the  range  of  temperature  to  which  it  is 
subjected  is  between  20°  F.  and  250°  F.  (the  length  of  the  girder  is  25  feet)  ? 

5.  Two  pendulums,  one  of  brass,  the  other  of  iron,  beat  seconds  when 
the  temperature  is  15°  C.  :  compare  the  number  of  beats  made  per  day  by 
each,  (a)  when  the  temperature  is  25°  C.  ;  (b)  when  it  is  —  10°  C. 

6.  In  an  experiment  illustrating  Lavoisier  and  Laplace's  method,  the 

.      ,GH         i 
ratio  of          = 


The  bar  H  K  is  copper  and  is  I  yard  long  at  o°  C.  When  the  bath 
is  filled  with  boiling  water  whose  temperature  is  100°,  the  distance  A  B  is 
found  to  be  35  divisions.  1000  divisions  of  this  scale  equal  18  inches. 
Find  the  expansion  of  copper. 

7.  The  density  of  gold  at  o°  C.  is  I9'2ii,  at  100°  C.  it  is  I9'I29'  find 
the  mean  coefficient  of  cubical  and  linear  expansion  of  this  metal. 


34  Heat 


CHAPTER   III. 
EXPANSION  OF  LIQUIDS. 

28.  Apparent  and  Absolute  Expansion.  —  In  measur- 
ing the  expansion  of  liquids  we  have  to  consider  the  expansion 
of  the  glass  envelope,  and  ordinarily  we  measure  the  apparent 
expansion  of  the  liquid.  The  student  should  repeat  the  ex- 
periment in  §  2  (Liquids). 

The  expansion  of  glass  makes  the  apparent  increase  in 
volume  less  than  the  real  increase  in  volume  of  the  liquid 

(§2). 

The  coefficient  of  apparent  expansion  of  a  liquid  in  a  given 
envelope  at  a  given  temperature,  is  the  ratio  of  the  apparent 
increase  in  volume  for  one  degree  to  the  original  volume. 

The  coefficient  of  absolute  expansion  at  any  temperature,  is 
the  ratio  of  the  absolute  increase  in  volume  for  one  degree  to 
the  original  volume. 

Let  A  be  the  true  dilatation  of  a  liquid  for  a  given  range 
of  temperature  ;  D,  the  apparent  dilatation  ;  G,  the  dilata- 
tion of  the  envelope  ;  V,  the  original  volume.  After  heating 
through  a  given  range  of  temperature,  since  D  is  the  apparent 
dilatation,  the  apparent  volume  is  V(i  +  D).  But  since  the 
envelope  has  dilated,  the  true  volume  of  every  apparent  unit 
of  volume  is  (i  -f  G)  ;  /.  the  true  volume  of  that  which  is 
apparently  V(i  +  D)  is  V(i  -f  D)(i  +  G).  But  this  equals 
V(i  -f  A),  since  A  is  the  true  dilatation. 


(i  +  A)  =  V(i 
(i+A)  =  (i 

,'.  A  =  D  -f  G  +  DG 


Expansion  of  Liquids  35 

The  whole  of  these  quantities  are  very  small,  and  the 
product  DG  can  be  neglected,  compared  with  A,  D,  or  G. 

.'.  A  =  D  +  G 

That  is,  the  real  increase  practically  equals  the  apparent 
increase  together  with  the  increase  in  volume  due  to  the  ex- 
pansion of  the  glass.  If  the  increase  of  temperature  be  i°, 
we  have — 

The  coefficient  of  absolute  expansion  of  a  liquid  equals 
the  apparent  expansion  of  the  liquid  together  with  the  co- 
efficient of  cubical  expansion  of  the  glass  enclosing  the 
liquid. 

29.  Weight  Thermometer. — The  apparent  expansion 
of  a  liquid  can  be  determined  by  a  weight  thermometer. 

A  glass  tube  about  6  inches  long,  with  an  internal  diameter 
of  \  inch,  is  closed  at  one  end ;  the  other  end  finely  drawn 
out  is  made  into  the 
shape  of  Fig.  25,  an 
is  left  open. 

The  apparatus  is 
weighed,  and  is  then 
rested  upon  a  conveni- 
ent support,  with  the 
open  end  dipping  into  mercury.  The  bulb  is  warmed.  On 
cooling,  mercury  is  forced  into  it ;  the  process  is  repeated  until 
the  thermometer  is  completely  filled  (the  process  of  filling  a 
thermometer  is  followed).  The  whole  is  allowed  to  cool  to 
the  temperature  of  the  room,  the  point  dipping  into  mercury. 
Any  adhering  particles  being  brushed  away,  it  is  weighed, 
and  the  temperature  is  taken.  Subtracting  the  weight  of 
the  glass,  we  obtain  the  weight  of  mercury  in  the  thermo- 
meter at  f. 

The  thermometer  is  now  placed  in  a  wire  cage,  care  being 
taken  that  no  mercury  escapes,  and  put  into  a  steam  bath. 
The  mercury  forced  out  as  the  temperature  rises  is  caught  in 
a  weighed  capsule.  When  no  more  mercury  escapes,  and  the 
temperature  is  steady,  the  temperature  of  the  bath  is  read, 
and  the  weight  thermometer  removed.  It  is  allowed  to  cool 


36  Heat 

to  the  temperature  of  the  room,  and  is  then  weighed  again. 
The  method  of  working  will  be  best  seen  from  an  example. 

Tube  +  mercury  at  15°  =  192-3  grams 
Empty  tube  =     15*6      „ 

.'.  weight  of  mercury  =  1767      „ 

Tube  +  mercury  at  100°  =190*0      „ 

Tube  =     15-6     „ 

/.  weight  of  mercury  =  174*4      „ 

174-4  grams  of  mercury  at  15°  expand  so  as  to  occupy  at 
100°  the  volume  of  1767  grams  at  15°. 

1767  -  174-4         2-3 
•'•  expansion  for  i  gram  =          ^^          =  — 

/.  coefficient  of  apparent  expan- 

sion of  mercury  in  glass  =  -       -  —  x  =  o'oooi^ 

17-4(100-15) 

In  this  particular  weight  thermometer  2-3  grams  are  ex- 
pelled when  the  temperature  rises  85°  ;  that  is,  0-027  gram 
is  expelled  when  the  temperature  rises  i°.  If,  then,  the 
thermometer  be  filled  when  the  temperature  of  the  room  is, 
say,  1  6°,  and  it  be  then  placed  in  an  enclosure  at  a  higher 
temperature,  and  we  collect  the  mercury  expelled  —  let  us  say 

i  -8  gram  —  then  the  temperature  of  the  enclosure  is  .  -  degrees 

above  the  room;  that  is,  the  temperature  is  (16  +  667)°  = 
82-7°  C.  The  origin  of  the  name  ""weight  thermometer"  is 
now  seen.  (See  also  Example  i,  p.  55.) 

In  a  general  example,  let  the  thermometer  be  first  weighed 


Let  V0  =  volume  of  the  thermometer  at  zero. 
D0  =  density  of  mercury  a.t  zero. 

M  =  mass  of  mercury  that  fills  the  thermometer  at  o°  C. 
;;/  =  mass  of  mercury  expelled  when  the  thermometer  is 

heated  to  /°. 
The  apparatus  at  f  contains  a  mass,  M  —  m.    :.  the  volume 

of  this  mass  at  zero  =  —  -  —  (since  M  =  V  x  D).      (§27.) 


Expansion  of  Liquids  37 

The  real  volume  of  this  mercury  at  f  -  —  =r  —  (i  +  A/), 

^0 

where  A  is  the  coefficient  of  absolute  expansion  of  mercury. 

The  thermometer  at  o°  contains  a  mass  of  mercury,  M, 
.".   its  volume  and   also  the  volume  of  the  glass  at  zero  is 

v^.  and  the  volume  of  the  glass  at  /°  =  fc-  (i  +  K/),  where 

Vg  UQ 

K  is  the  coefficient  of  cubical  expansion  of  glass.     This  equals 
the  real  volume  occupied  by  M  —  m  of  the  mercury  at  f 


Also,  if  d  =  coefficient  of  apparent  expansion  of  mercury  in 
glass— 

(i+A/)  =  ('+<fl)(i  +  K/)   (§28.) 
.'.  (M  -  m)(i  +dt)  =  M 

m 


~  (M  -  m)t 

In  order  to  use  the  instrument  as  a  thermometer,  a  careful 
determination  should  be  made  between  o°  C.  and  100°  C. ;  the 
amount  expelled,  m,  divided  by  100  (call  this  //,)  will  give  the 
weight  expelled  for  a  rise  of  i°.  If  in  any  other  experiment 

n  grams  be  expelled,  then       =  the  rise   in   temperature   in 

degrees. 

We  assume  that  the  expansion  of  mercury  and  glass  is 
uniform  from  o°  C.  to  100°  C. ;  that  is,  the  expansion  for  i°  is 

-  of  the  expansion  from  o°  to  100°  C. 
100 

30.   The   Absolute   Expansion   of   Mercury. — The 

results  obtained  by  Dulong  and  Petit  (§  31)  are  given  in  the 
first  column  ;  the  approximations  of  Regnault  (whose  method 
is  given  in  §  32)  are  given  in  the  second  column.  The  student 
will  notice  that  the  coefficient  increases  with  the  temperature. 


38  Heat 

MEAN  COEFFICIENTS  OF  THE  ABSOLUTE  EXPANSION  OF 
MERCURY. 


Between  o°  and  iooc 


„        o°    „    300° 

The  coefficient  of  apparent  expansion  of  mercury  in  glass 
is  generally  given  as  -gTg^.  following  Dulong  and  Petit.     The 

fraction  will  vary  with  the  kind  of  glass  used,  and  should  be 
determined  for  each  instrument. 

(a)  (!>)  (c) 


Dulong  and  Petit. 

I 

Regnault. 

I 

5550 

I 

5508 

I 

5425 
I 

5425 

I 

5360 

5300 

Coefficient  of  abso- 
lute expansion  of 
mercury 


(Coefficient  of  appa-  j     (Coefficient     of 
=  |    rent  expansion  of  >  +  ]  cubical  expan- 
(    mercury  j     (  sion  of  glass. 

We  can  determine  (b)  (§  29).  If  (a)  were  known,  we  have 
a  ready  method  of  calculating  (c).  • 

For  example,  taking  Dulong  and  Petit's  results,  (c)  =  - 

555° 

—  7-Q-  =  -o o '0000258.     For  ordinary  chemical  glass 

6480      30700 

0-0000254  is  a  better  value. 

These  determined,  we  could  substitute  any  other  liquid,  say 
glycerine  or  water,  in  the  weight  thermometer,  by  experi- 
ment find  (b)  for  the  liquid,  then  (c)  being  known,  (a)  for 
the  liquid  would  be  known. 

The  important  matter  is  to  find  the  coefficient  of  absolute 
expansion  for  some  liquid  by  some  independent  method. 
Mercury,  seeing  that  it  can  be  obtained  pure,  that  it  does  not 
at  ordinary  temperatures  evaporate,  and  that  it  is  used  in  many 
experiments,  has  been  subjected  to  careful  examination.  The 
method  is  long,  and  demands  special  precautions;  the  principle, 
however,  is  readily  grasped. 


Expansion  of  Liquids 


39 


31.  Dulong  and  Petit's  Method. — If  we  take  a  U  tube 
(Fig.  26),  place  it  vertically,  pour  mercury  sufficient  to  rise 
slightly  in  each  limb,  and  then  pour  any  liquid  into  one  limb  ; 
then,  if  we  draw  a  horizontal  plane,  B  C,  through  the  surface 
of  separation,  B,  we  know,  by  the  hydrostatical  principle,  when 
all  is  in  equilibrium,  that  the  heights  of  the  liquids  above 
the  surface  of  contact  are  in  the  inverse  ratio  of  their  densities. 
For  example,  if  B  C  D  be  mercury,  and  B  A  water,  B  A  in  the 
tube  ;/  will  be  about  13^  times  the  height  of  C  D  in  the  tube  m  ; 


FIG.  26. 


FIG.  27. 


that  is,  the  density  of  mercury  is  about  13  J  times  the  density 
of  water.  This  ratio  of  their  densities  is  independent  of  the 
diameter  of  the  vessel. 

It  was  by  applying  this  principle  that  the  first  determinations 
of  the  absolute  expansion  of  mercury  were  made  by  Dulong 
and  Petit  Vertical  tubes,  A  B,  C  D,  were  joined  by  a  fine 
horizontal  tube,  B  D  (Fig.  27).  Mercury  was  poured  in,  and 
of  course  stood  at  the  same  height  in  each  tube.  A  B  was 
surrounded  with  melting  ice,  and  C  D,  we  shall  imagine, 
with  a  steam  bath  (it  was  really  a  bath  of  oil).  Then,  B  D 
being  in  equilibrium,  the  pressure  at  D,  due  to  the  column 
D  C  and  the  pressure  of  the  atmosphere  equalled  the  pressure 
at  B,  due  to  the  column  A  B  and  the  pressure  of  the  atmo- 


40  Heat 

sphere.  The  pressure  of  each  column  is  due  to  the  vertical 
height  and  the  density  of  the  liquid.  If  D0  and  D100  represent 
the  density  of  mercury  at  o°  and  100°,  and  h  and  H  the  height 
of  the  mercury  in  each  tube  above  B  D,  then — 

D0>&  =  D100H  (i.) 

And  if  V0  and  V100  be  the  volumes  of  equal  masses  at  the 
two  temperatures  (§  27),  then — 

D0V0  =  D100V100  (ii.) 


v0 


VQ(I 


m  T 


H 
h 


A  being  the  coefficient  of  the  absolute  expansion  of  mercury. 

TT     __      7. 

H  and  h  were  carefully  measured,  and  thus  A  =  -  - 

h.  100 

was  determined. 

The  complete  apparatus  is  shown  in  Fig.  28.     A  and  B 


FIG.  28. 


are  the  two  tubes ;  E  contains  oil  heated  by  the  furnace ;  T  R 
is  an  air -thermometer ;   and  P  C  a  weight  thermometer  for 


Expansion  of  Liquids  41 

determining  the  temperature  of  E.  The  vessel  D  is  filled  with 
melting  ice ;  m  n  are  levels  by  means  of  which  the  stand  M  K 
is  levelled. 

The  difficulty  of  keeping  the  mercury  at  a  constant 
temperature  and  at  a  constant  height,  and  of  preventing  the 
transference  of  heat  from  B  to  C,  has  led  to  the  following 
modifications  : — 

32.  Regnault's  Method.— A  B,  C  D  are  vertical  tubes, 
continued  above  A  and  C,  and  open  at  the  top  (Fig.  29).  They 
are  joined  by  a  hori- 
zontal tube,  A  C. 
B  E,  D  F  are  hori- 
zontal tubes,  into 
which  vertical  tubes, 
E  K,  F  L,  are  inserted. 
These  are  joined  to- 
gether at  the  top  by 
a  tube,  K  L.  Into 
K  L  opens  a  tube, 
m  u,  connected  with 
a  spherical  vessel,  P, 
of  compressed  air. 

A  B  is  surrounded 


FIG.  29. 


with  a  bath  of  boiling  oil;  while  into  the  bath  C  D  water 
enters  at  the  bottom  and  escapes  at  the  top.  The  overflow 
water  runs  over  K  L,  L  F,  and  t^he  parts  of  E  B,  F  D  not 
exposed  to  the  baths,  and  keeps  them  at  a  constant  tempera- 
ture. Thermometers  (not  shown)  are  inserted  in  the  baths,  so 
that  their  temperatures  can  be  accurately  determined.  Mercury 
is  poured  into  A  B,  C  D.  The  two  parts  are  prevented  from 
mixing  by  the  air  in  K  L.  P  is  kept  at  a  constant  temperature 
by  being  surrounded  by  water,  and  more  air  can  be  forced  in 
through  n  if  necessary. 

The  air  is  compressed  until  the  mercury  is  above  the  level 
A  C.  Seeing  that  A  and  C  are  in  communication,  the  total 
pressure  of  mercury  and  atmosphere  above  A  must  equal  that 
above  C.  The  distances  above  A  and  C  need  not  therefore  be 
measured.  The  mercury  fills  A  B  E  to  K  on  the  left  side,  and 


42  Heat 

C  D  F  to  L  on  the  right  side.     Suppose  all  the  temperatures 
steady,  then  the  following  observations  are  taken  : — 

H   =  the  height  of  A  above  the  tube  B  E. 
h    =      „          „     K        „          „      BE. 

Hr\  T-V    Tf 

l  —         jj  •>•>        ^  11  »        -M  *• 

//i    =      „  „     L         „  „      DF. 

The  pressure  of  the  mercury  at  K  is  equal  to  the  difference 
of  the  pressure  due  to  the  height  of  mercury  H  at  T°  and  h 
at  /°.  Similarly,  the  pressure  at  L  equals  the  difference  due  to 
the  height  H!  at  t°  and  the  height  hv  at  /°.  These  pressures 
are  equal,  being  both  equal  to  the  pressure  of  the  compressed 
air  in  M. 

If  A  be  the   coefficient  of  absolute   expansion,  we  have 

length  H  at  T°=  -   ^—  at  o°  (see  §  39).    Reducing,  similarly, 

all  to  o°— 

h  /I-, 


i+AT     i+A/~ 


If  we  assume  that  the  coefficient  of  expansion  can  be  ex- 
pressed by  a  simple  fraction,  A  is  the  only  unknown  quantity. 

But  Regnault  made  no  such  assumption.  It  was  one  of 
the  objects  of  his  research  to  discover  whether  the  expansion 
of  mercury  was  uniform  ..or  not.  t  would  not  be  a  high 
temperature,  and  therefore  A/  would  be  small  compared  with 

AT.     He  therefore  assumed  --  or  other  suitable  value  from 

Dulong  and  Petit's  tables  (p.  38),  and  calculated  A/;  sub- 
stituting this  value  in  the  formula,  he  obtained  an  approximate 
value  of  AT  as  a  whole,  that  is,  the  total  dilatation.  From 
this  value  of  AT  he  determined  the  mean  coefficient  A,  cal- 
culated A/,  and  used  this  in  the  formula  for  determining  the 
final  value  of  AT.  (See  Worked  Example  3,  p.  56.) 

33.  Formulae  and  Results.  —  This  is  a  convenient  place 
to  note  that  the  coefficients  of  expansion  are  not  necessarily 
of  the  simple  form  we  have  assumed;  that,  for  instance, 


Expansion  of  Liquids  43 

V,  =  V0(i  4-  K/)  does  not  really  represent  the  truth.  If  the 
coefficient  of  expansion  were  constant,  and  always  equal  to 
K,  then  the  total  dilatation  for  unit  volume  for  any  range  of 
temperature,  /,  is  K£ 

Suppose  we  mark  off  on  O  X  temperatures,  and  along  O  Y 
volumes  (Fig.  30).  The  position  O  on  O  Y  represents  10,000 
volumes.  Let  us  take  the  coefficient  for  mercury  to  be  the 
mean  between  o°  C.  and  100°  C.,  that  is,  0*00018153;  then  a 
is  the  position  for  100°,  the  distance  from  the  temperature  line 
to  a  being  181*5  >  f°r  200°  ft  w^  be  3^3'ij  represented  by  b\ 
for  300°  it  will  bs  544*6,  represented  by  c.  Joining  the  points 
O  a  b  c,  we  form  the  straight  dotted  line. 

Regnault's  investigations  showed  that  the  relation  was 
more  accurately  expressed  by  a  curve,  which  was  convex 
towards  O  X ;  that  the  actual  volumes  above  100°  C.  were 
greater  than  is  indicated  by  the  dotted  line  O  a  b  c,  while  below 
1 00°  they  were  less.  The  continuous  curve  represents  as 
carefully  as  is  possible  on  this  scale  some  of  the  results  of 
Regnault's  research. 

The  dilatation  for  /°  (A/),  instead  of  being  represented 
as  K/,  is  more  accurately  represented  by  three  terms ;  thus — 
A/  =  at  +  bt-  4-  ct* 

The  constants  have  been  determined  that  give  accurate 
results  between  certain  limits.  The  mean  dilatation  for  one 
degree  between  these  limits  will  be — 

—  =  a  -f  bt  4-  ct- 

a  and  b  and  c  are  all  three  positive ;  and  we  see  that  the  mean 
coefficient  of  expansion  increases  with  the  temperature. 

A  useful  approximation  for  mercury  is  given  by  A/  =  at 
4-  £/2,  where — 

a  —  o'oooiSoi 


=  0'OOOOOOO2 
o 


Thus  unit  volume  at  o°  would  become  at  50° — 

=  i  4-  at  4-  btz  -  i  4-  0*0001801  X  50  +  0-00000002  X  (so2) 

=  i  4-  0-009005  +  0-00005  =  1-009055 

Regnault's  experiments  gave  1-009013. 


44 
y 


/0.50Q. 


10.200 


a 


TEMPERATURE   , 

10  o_c 20  ore 


FTG.  30. 


Expansion  of  Liquids 


45 


In  finding  the  mean  coefficient  between  two  temperatures, 
we  find  the  ratio  between  the  increase  in  volume  and  the 
increase  in  temperature. 

The  true  coefficient  of  expansion  at  any  given  temperature 
is  found  by  taking  a  very  small  increase  in  volume,  called  dv9 
and  dividing  it  by  the 
small  increase  in  tem- 
perature, say  dt  ;  that 
is,  we  measure  the  rate 
of  change  of  the  volume 
as  the  temperature 
changes,  so  that  the 

dv 

coefficient   c  =  —  .    In 

Fig.  31,  when  tempera- 
ture increases  from  /  to 
/!,  the  volume  increases 


FIG.  31. 


QR 


If 


by  Q  R.     Then  the  coefficient  is  ~-j  =  -~  =  tan   6. 

*i  —  t        i  Is. 

OR 
/!  be  very  near  /,  so  that  Q  moves  to  P,  -p^  becomes  the  tan- 

gent of  the  curve  at   the   point   P,  and   measures  the  true 
coefficient  of  expansion  at  /°. 

The  form  of  the  curve  (Fig.  30)  shows  that  the  true  co- 
efficient will  increase  as  the  temperature  is  raised. 

TABLE  OF  THE  DILATATIONS  OF  MERCURY. 


mperature 

Volume  at  each 
temperature  f. 

Mean  coefficient  of 
expansion  from 
o°  to  *°. 

True  coefficient  of 
expansion  at  f. 

0°       ... 

I  "OOOOOO 

O'OOOOOOOO 

O*OOOI7905 

20°       ... 

I'003590 

0*OOOI7Q5I 

0*00018001 

40°     ... 

I*OO72IO 

O  '00018002 

O*OOOl8lO2 

60°   ... 

I*OI083I        ... 

0*00018052 

O*OOOl82O3 

80°   ... 

1*014482        ... 

0*00018102 

0*000l8304 

100°       ... 

I-OI8I53        ... 

0-00018153    ... 

0-OOOI8405 

200°       ... 

1*036811 

0*00018405 

0*00018909 

300°       ... 

1*055973       ... 

0*00018658 

0*OOOI94I4 

The  increase  in  the  true  coefficient  of  expansion,  and  the 


46 


Heat 


difference  between  the  true  and  mean  coefficients,  are  very 
small,  and  probably  the  student  will  not  be  called  upon  in 
any  ordinary  experiments  or  calculations  to  use  such  exact 
figures.  The  table  is  given  to  illustrate  the  present  section, 
and  to  accentuate  the  fact  that,  while  for  ordinary  purposes  it 
may  be  sufficient  to  use  a  simple  form  for  the  coefficient  of 
expansion  of  mercury  (or  other  substance),  the  physical  facts 
may  not  be  as  simple  as  the  form  would  suggest,  and  that 
it  is  only  by  laborious  experiments  that  the  actual  truth  can 
be  reached.  Later  researches  have  slightly  altered  the  figures 
given,  and  probably  as  the  skill  of  experimenters  increases 
and  apparatus  is  improved,  further  slight  alterations  will  be 
needed.  These  remarks  will  be  further  illustrated  throughout 
the  book,  notably  when  we  treat  of  what  to  the  beginner  are 
the  simple  laws  of  Boyle  and  Charles.  Analogous  results  are 
obtained  for  other  liquids.  The  case  of  water  will  be  treated 

in  §  35- 

34.  Maximum  Density  of  Water. — The  experiments 

of  Hope  showed  that,  if  water 
be  gradually  cooled  down  to 
the  freezing  point,  its  volume 
.does  not  decrease  continu- 
ously ;  that  is,  that  its  maxi- 
mum density  is  not,  as  we 
might  expect,  at  o°  C. 

A  tall  jar  is  fitted  with  two 
thermometers  in  a  horizontal 
position  (Fig.  32).  The  jar 
is  filled  with  water  at  o°,  and 
is  then  brought  into  a  room 
at,  say,  10°.  The  water  at 
the  side  becomes  heated,  and 
rises,  the  lower  thermometer 
indicating  a  lower  tempera- 
ture than  the  upper  until  the 


FIG.  32. 


lower  registers  4°  C.  The  upper  falls  to  4°,  and  gradually 
to  o°,  while  the  lower  remains  at  4°.  Evidently  the  density  of 
water  at  4°  is  greater  than  it  is  at  o°. 


Expansion  of  Liquids  47 

The  converse  experiment  is  made  by  filling  the  jar  with 
water  at  6°  or  7°,  and  then  taking  it  into  a  room  at  o°.  At 
first  the  lower  thermometer  indicates  a  lower  temperature 
until  the  whole  sinks  to  4°.  The  lower  then  remains  at  4°, 
while  the  upper  falls  to  freezing  point 

The  jacket  of  ice  (Fig.  32)  is  placed  round  for  the  purpose 
of  a  lecture  experiment.  The  cooled  water  sinks  until  4°  is 
reached,  then  the  water  at  4°  remains  at  the  bottom,  and  the 
water  at  3°,  2°,  i°,  o°  rises,  and  the  upper  thermometer  falls  to 
freezing  point 

In  nature  the  result  is  that  the  surface  water  of  ponds  and 
lakes  sinks  until  the  temperature  of  the  whole  is  at  4°.  The 
next  layer,  cooled  below  4°  C.,  floats  until  its  temperature  sinks 
to  o°.  It  then  freezes,  and  the  density  of  ice  being  less  than 
water,  the  ice  floats,  and  thus  the  general  temperature  of  the 
latter  is  prevented  falling  below  4°,  and  fishes  and  plants  that 
would  otherwise  perish  are  preserved. 

35.  Absolute  Expansion  of  Water. — We  have  seen 
(§29)  that  the  weight  thermometer  can  be  used  for  determining 
the  expansion  of  water  (or  other  liquid).  Knowing  the  absolute 
expansion  of  mercury,  we  can  measure  the  apparent  expansion 
when  enclosed  in  a  particular  kind  of  glass,  and  deduce  the 
expansion  of  the  glass.  Filling  the  thermometer  with  water, 
we  can  determine  the  apparent  expansion  of  water,  and  then 
deduce  the  absolute  expansion  of  water  (§  30).  In  practice  this 
method  is  found  to  be  defective.  The  density  of  water  is  only 
about  y^  that  of  mercury,  and  the  amounts  expelled  are  there- 
fore small ;  errors  are  also  introduced  by  evaporation. 

Thermometer  Method. — A  better  plan  is  to  use  a  large  gradu- 
ated thermometer.  The  coefficient  of  expansion  of  the  kind 
of  glass  is  determined  by  making  part  into  a  weight  thermo- 
meter. The  tube  is  calibrated  carefully  (§  6),  and  the  volume 
of  the  bulb  is  determined  by  filling  the  bulb  and  a  certain 
number  of  divisions  with  mercury,  weighing,  and  deducting  the 
weight  of  the  glass.  This  gives  a  mass,  M,  of  mercury  ;  then, 
if  D  be  the  density  of  mercury  at  the  given  temperature,  we 

have  V,  =  — ;  that  is,  the  volume  is  known  at  /°.     Also  V0  = 


48  Heat 

y 

*  .,  where  c  is  the  dilatation  of  glass  from  o  to  t  degrees. 

We  next  determine  the  volume  of  a  division  of  the  stem  by 
introducing  a  thread  of  mercury,  observing  its  length  and  the 
number  of  divisions  it  covers.  If  m  be  the  weight  of  the 
thread,  and  n  the  number  of  divisions,  and  /  the  temperature — 

m  vt 

-=,,;also*.  =  — - 

V  v  m 


n      n(i  +  a)      «D(i  -f  ct) 

=  volume  of  each  division  at  o°,  and  the  volume  at  any  other 
temperature  is  easily  calculated.  (D  =  density  at  /°.) 

The  whole  of  the  mercury  is  expelled,  and  the  liquid 
introduced ;  its  volume  (vt)  at  f  is  observed.  It  is  then  heated 
to  various  temperatures,  the  new  volume  noticed,  and  thus  the 
apparent  expansion  between  o°  and  a  series  of  temperatures 
/i,  4,  etc.,  is  determined.  The  absolute  expansion  is  then 
obtained,  knowing  the  expansion  of  the  glass.  The  expansion 
of  water  has  been  carefully  studied  by  this  method. 

If  the  water  be  thoroughly  freed  from  air,  it  has  the 
property  of  remaining  liquid  at  temperatures  down  to  —  20°  C., 
so  that  the  dilatation  can  be  examined  down  to  this  tempe- 
rature. The  maximum  density  of  water  is  found  to  be 
at  4°  C. 

The  Areomelric  Method. — This  method  was  first  used  by 
Matthiesson  to  determine  the  absolute  expansion  of  water. 
A  delicate  balance  is  required.  A  long  fine  platinum  wire 
passes  from  the  end  of  one  arm  through  a  hole  in  the  bottom 
of  the  case,  and  dips  into  a  bath  of  pure  water,  which  can  be 
kept  at  a  constant  temperature.  The  steps  are  as  follows  : — 

(1)  A  piece  of  glass  rod,  whose  linear  expansion  has  been 
determined  accurately  by  the  ordinary  method,  is  weighed  in 
air.     Let  its  mass  be  m. 

(2)  The  solid  is  attached  to  the  platinum  wire,  and  weighed 
when  immersed  in  pure  water  at  two  definite  temperatures. 
Let  us  take  the  temperatures  at  o°  and   100°;   and   let  the 
weights  be  m§  and  mm. 


Expansion  of  Liquids  49 

Then  /x0  =  ;;/  —  w0  =  weight  of  water  displaced  at  o°  C. 
An-J  /AJOO  =  m  —  mm  =          „  „  „  ioou  C. 

(3)  Let  c  be  the  linear  coefficient  of  expansion  of  glass. 
Then  $c  =  K  =  coefficient  of  cubical  expansion. 

Let  a  be  the  mean  coefficient  of  expansion  of  water  be- 
tween the  two  temperatures. 

—  =  spec,  gravity  of  the  solid  when  solid  and  water  are  at  oc 

;;/ 

—  >»  »  »  100° 

If  D  and  d  represent  the  density  of  glass  and  water,  then 
-r  and  —^  also  equal  the  specific  gravity  of  the  solid  under 

"0  "100 

the  two  conditions. 

.*.  —  =  -A  and  -  -  =  -7^ 

/*•()  ^0  A^lOO  ^*100 

b. 


Also  D'»  =  i  +  i°ooK  and  '»  =  IT 


m        D100  _  i  +  iooK  _  D0 

f*100    ~     ^100  "     </0         I   +   IOOK 


m      i  4-  iooa 
~  P.Q  '  i  4-  iooK 

Mo 

iooa  =  —  (i  4-  iooK)  —  i 


All  the  quantities  are  known  save 

If  the  two  temperatures  be  /x  and  /2,  the  student  will  be 
able  to  show  that  — 


i  4-  /^      /x.,     i  4  4K 

(See  Worked  Examples,  No.  4,  p.  56.) 

36.  Direct  Measurement  of  Cubical  Expansion  of 
Solids. — The  cubical  dilatation  of  solids  has  been  determined 
in  §  25,  from  the  linear  dilatation.  If  we  know  the  absolute 

£ 


£O  Heat 

expansion  of  any  liquid,  we  can  determine  the  cubical  expan- 
sion of  a  solid  direct  by  weighing  it  in  air  and  in  the  liquid. 

If,  for  example,  by  any  other  method  (Regnault's)  we  know 
accurately  the  expansion  of  water,  then  we  can  use  the  areo- 
meter to  determine  the  cubical  expansion  of  solids ;  that  is, 
in  the  last  formula  all  the  quantities  are  known  save  K. 

TABLE. 

The  following  represent  some  mean  values  of  the  volume 
occupied  by  water,  whose  volume  is  unity  at  4°  C.  :— 
—  2°.   1-000308  7°.   i -000066 

o°.   1*000130  8°.   i -000166 

i°.  1*000072  10°.   1-000260 

2°.  1-000032  15°.   1-000855 

3°.   i  "000008  20°.  1*001746 

4°.   1*000000  40°.   1*007695 

5°.   i  "000008  60°.   1*016919 

6°.   1*000030  100°.   1*043130 

By  either  of  the  methods  in  §  35  it  is  found  that  a  given 
mass  of  water  occupies  a  minimum  volume  at  a  temperature 
that  is  practically  4°  C.  Dividing  the  volume  at  any  temperature 
by  the  volume  at  4°,  we  obtain  the  volume  that  unit  volume 
occupies  at  various  temperatures.  An  examination  of  the 
table  shows  that  the  rate  at  which  the  volume  increases  from 
4°  C.  increases  with  the  temperature;  that  is,  the  mean 
coefficient  of  expansion  of  water,  and  also  the  true  coefficient 
at  any  temperature,  increase,  with  the  temperature. 

Following  the  plan  in  §  33  for  mercury,  empirical  formulae 
have  been  devised  for  water  and  other  liquids.  Generally 
the  form  is — 

V,  =  V0(i  4-  at  -f  bt*  +  a3} 
where  a,  fr,  and  c  have  to  be  determined  for  each  liquid. 

Mendeleeff  has  shown  that  if  the  density  of  water  at  4°  C. 
under  a  pressure  of  i  atmosphere  be  unity,  then  the  density 
at  any  other  temperature — 


where  <£(/)  =  125-78  4-  I'isS/  -  0-0019/2. 


Expansion  of  Liquids 


1000(137-17) 


This    gives   tbe^  density  at    10°   C.  =  i  — 
-  i  —  0*000262  =  0-999738. 

.*.  unit  volume  at  4°  C.  =  i  -f-  0-999738  =  i '000262 

The  density  at  100°  C.  by  the  same  formulae  becomes 
0-958595  and  volume  1-043194. 

The  formula  gives  good  results  between  — 10°  C.  and 
100°  C. 

37.  Other  Liquids. — The  coefficients  of  expansion 
generally  increase  with  the  temperature,  and  liquids  near  the 
boiling  point  show  a  marked  increase. 

This  is  shown  in  the  following  table.1  The  temperature 
after  the  liquid  is  that  of  its  boiling  point : — 

VOLUMES. 


Temperature. 

Alcohol. 

78°. 

Ether. 
35°. 

Bisulphide 
of  carbon. 
46°. 

Water. 

100°. 

Air. 

O° 

10 

ro 

I'O 

•o 

I  XI 

10° 

1-01050 

1-01518 

I-OII56 

•00013 

1-0367 

20° 

1-02128 

1-03122 

I-02350 

•00l6l 

1-0734 

30° 

1-03242 

i  -04829 

I'03594 

•00414 

I'HOI 

40° 

I  -04404 

1-06654* 

I-0490I 

•00756 

1-1348 

*  Under  increased  pressure. 

Air  is  added  for  reference. 

By  subjecting  the  liquids  to  pressure,  they  can  be  kept  in 
the  liquid  form  at  temperatures  above  their  ordinary  boiling 
points,  which  are  determined  under  normal  atmospheric 
pressure.  The  coefficients  are  then  found  to  increase  rapidly. 
Thus  liquid  carbonic  acid  between  o°  and  30°  has  a  coefficient 
four  times  greater  than  that  of  air.  Water  at  180°  C.  has 
a  coefficient  nearly  half  that  of  air,  while  the  coefficient  of 
alcohol  at  160°  C.  is  nearly  five  times  that  of  air. 

38.  Graham's  Pendulum. — In  its  simplest  form  it 
consists  of  a  steel  (or  other  metal)  rod  attached  to  a  cylindrical 
glass  vessel  containing  mercury  for  a  bob.  With  a  rise  of 

1  Everett's  "Physical  Constants." 


$2  Heat 

temperature,  the  steel  expands,  and  lengthens  the  pendulum  ; 
the  mercury  also  expands,  and  thus  raises  the  centre  of  gravity 
of  its  mass. 
Between  o°  and  40°  — 

The  linear  coefficient  of  expansion  of  steel  is  0*000012 

The  cubical        ,,  „       mercury  „  0*000180 

If,  therefore,  the  rod  be  attached  to  the  base  of  the  cylinder 
the  pendulum  will  compensate,  neglecting  small  corrections 
when  the  length  of  the  steel  rod  is  to  the  length  of  the  column 
of  mercury  as  180  :  12  =  15  :  i. 

39.  Barometric  Corrections.  —  The  pressure  of  the 
atmosphere  is  measured  by  the  pressure  of  a  column  of  mer- 
cury of  an  observed  height  on  a  given  area;  the  normal 
pressure  is  equal  to  that  of  a  column  of  mercury  760  mm. 
high  per  unit  of  area,  say  per  square  millimetre.  To  determine 
the  pressure  we  determine  the  weight  of  760  cubic  mm.  of 
mercury:  this  equals  760  X  weight  of  i  cubic  mm. 

As  the  density  of  mercury  changes  with  the  temperature, 
it  is  necessary  to  have  some  standard  temperature;  this 
temperature  on  the  Centigrade  scale  is  o°  C.  If,  therefore, 
the  height  760  mm.  be  observed  at  15°,  the  true  height  will 
be  the  height  this  column  would'  assume  if  the  temperature 
were  o°. 

(i)  Barometric  readings  will  not  usually  be  taken  at  a 
higher  temperature  than  40°  C.,  therefore  we  may  take  the 
coefficient  of  mercury  as  o  'ooo  1  8. 

If  L  be  the  length  of  the  column,  then  — 

Lt  =  L0(i  +  o'oooiS/) 


LQ  is  called  the  true  reading  of  the  barometer. 
L0  =  L,  —  o'oooiS/  .  L, 

Seeing  that  0*00018  is  small,  and  the  barometer  will  not 
Vary  in  these  latitudes  much  from  760  mm.,  we  may  write  — 

L0  =  L,  —  o'oooiS  X  760  X  / 
=  L,-  o-i37/ 


Expansion  of  Liquids  53 

that  is,  from  the  observed  reading  in  millimetres  subtract  0-137 
X  temperature. 

Thus  if  the  barometer  reading  be  763  -6  at  15°,  the  true 
reading 

=  763-6  -  0-137  x  15 

=  763-6  -  2-055 

=  761-545  approximately 

The  true  value  would  be  763*6  —  2*061  =  761*539. 

(2)  Not  only  has  the  column  of  mercury  expanded,  but 
the  scale  (generally  brass)  on  which  the  graduations  are  marked 
has  also  expanded,  and  therefore  the  true  reading  will  be 
greater  than  the  observed  reading.  The  scale  is  generally 
made  correct  at  o°  C.  Since  the  expansion  of  mercury  is 
to  the  linear  expansion  of  brass  as  18  :  1*9,  this  correction 
will  be  less  than  that  due  to  the  expansion  of  mercury. 

If  /  be  the  observed  length  of  the  scale,  the  true  length 
will  be— 

/(i  +  o'ooooi9/)  =  /+  o'ooooig// 

Again,  taking  a  mean  value  for  /  =  760 


In  the  above  example  the  correction  for  the  brass  scale 
will  be  — 

0*01444  x  15  =  o  217 

/.  making  both  corrections,  the  true  reading  is  — 
763-6  —  2-05  -f-  0-22  =  761*77  mm. 

Generally,  if  a  barometer  scale  be  correct  at  o°,  any  observed 
reading  /  in  mm.  at  f  needs  two  corrections  :  (a)  mercury 
-K//;  (b)  scale,  +  /?//. 

/.  true  reading  =  /  -  /(K  -  /?)/ 
For  brass  and  mercury  this  becomes  — 

/  —  /(o'oooiS  — 
—  I  —  /(o*oooi6i)/ 

or,  taking  a  mean  value  of  /  =  760 

=  /  —   O  I22/ 


54  Heat 

Sometimes  the  graduations  are  on  the  glass  itself.  In  this 
case  the  true  reading 

=  /  —  /(o'oooiS  —  0*0000085 
=  /—  7(0-00017 15) 
=  /—  760(0*00017 15) 

—  I  —  Q'I2Ot 

Other  corrections  are  (a)  those  due  to  capillarity.  This 
requires  a  small  quantity,  depending  upon  the  diameter  of  the 
tube,  to  be  added  to  the  result.  If  the  tube  be  5  mm.  in 
diameter,  then  1*54  mm.  must  be  added  to  the  observed  height; 
if  3  mm.,  then  1*22  mm.  The  exact  figures  can  be  taken  from 
proper  tables. 

(b)  The  student  will  see,  after  §  97,  that  the  pressure  of 
mercury  vapour  is  appreciable,  and  will  depress  the  column, 
and  thus  a  further  amount  must  be  added. 

Having  made  the  four  corrections  :  (i)  mercury,  (2)  scale, 
(3)  capillarity,  (4)  vapour  pressure,  we  obtain  the  true  height 
of  the  barometer.  This  equals  the  pressure  of  the  atmosphere 
near  the  barometer  when  the  reading  is  taken. 

40.  Pressure  of  the  Atmosphere. — If,  after  all  the 
above  corrections  have  been  made,  the  true  height  is  750  mm., 
then  the  pressure  of  the  atmosphere  is  equivalent  to  the 
pressure  of  75  cubic  cm.  per  square  centimetre. 

The  density  of  mercury  at  o°  =  13-  5 96. 

/.   the  pressure   of   the   atmosphere   per   square   centimetre 
equals  the  weight  of  a  volume  of  mercury  whose  mass  is 
75  X  13*596  grams  =  1019-7  grams  per  square  centimetre 
This  will  depend  upon  g,  the  acceleration  due  to  gravity. 
For  purposes  of  comparison  the  standard  value  of  the  accelera- 
tion due  to  gravity  (g0)  is  taken  at  the  sea-level  in  latitude  45  N. 
If,  after  the  above  corrections,  the  height  of  the  barometer  is 
/!  in  the  latitude  <£,  and  at  a  height  h  in  metres  above  the  sea- 
level,  and  gl  be  the  acceleration  due  to  gravity  under  these 
conditions,  it  can  be  shown  that— 

^  =    I    —  0*0026  COS  2</>  —  0'OOOOOO2/Z 


Expansion  of  Liquids  55 

then  if  /0  be  the  reading  for  g  =  g<>  — 


(l  —   0*0026  COS  <£  —  0*0000002^) 
« 

The  last  part  of  the  expression  can  generally  be  neglected 
unless  it  be  greater  than  1000  metres,  i.e.  than  3281  feet. 


WORKED  EXAMPLES. 

I.  A  weight  thermometer  when  empty  weighs  11-5  grams;  filled  with 
mercury  at  13°  C.,  it  weighs  141*56  grams.  After  heating  to  100°  the 
weight  is  I39'86  grams.  The  cubical  expansion  of  glass  has  been  found  to 
be  0*0000264.  Find  (i)  the  apparent  expansion  of  mercury  in  glass; 
(2)  the  absolute  expansion  of  mercury  in  glass. 

The  vessel  contains  130-06  grams  at  13°  and  128-36  grams  at  100°. 

.*.  1*7  gram  has  been  expelled 

The  apparent  expansion  can  be  calculated  from  the  formula  in  §  28,  or 
the  result  can  be  obtained  thus  — 

Since  we  are  not  concerned  with  the  expansion  of  glass,  we  may  regard 
the  glass  as  being  inexpansible.  The  thermometer  is  full  at  100°  and  at  13°  ; 
that  is,  130*06  grams  at  13°  and  128-36  grams  at  ioo°  each  occupy  the 
same  volume,  V. 

V 


/.  dilatation  of  unit  mass  = 
/.  mean  coefficient  of  apparent  expansion  of  mercury  between  13°  and  1  00° 


increase  in  unit  volume  _     \  128  -36  x  1  30*06  /  __          1*7 
unit  volume  X  87  I  ~"  128*36  X  87 

V  *  13^06  X  87 
=  0*000152 

,*.  coefficient  of  absolute   expansion  =  coefficient  of  apparent  expansion 
+  expansion  of  glass  =  0-000152  +  0*0000264  =  0-000178 

2.  The  coefficient  of  absolute  (cubic)  expansion  of  mercury  is  0*00018  ; 
the  coefficient  of  linear  expansion  of  glass  is  0*000008.  Mercury  is  placed 
in  a  graduated  tube,  and  occupies  100  divisions  of  the  tube.  Through 
how  many  degrees  of  the  tube  must  the  temperature  be  raised  to  cause  the 
mercury  to  occupy  101  divisions?  (London  Matric.,  1883.) 


56  Heat 

If  t  be  the  number  of  degrees,  then  the  length  of  the  mercury  column 
becomes  100(1  +o-oooi8/). 

These  equal  101  divisions  of  the  glass  after  expansion. 

100(1  +  o*oooi8/) 
.*.  I  glass  division  =  - 

I  glass  division  becomes  (i  +  o-oooooS/)  divisions  at  t° 
/.  100(1  +  0-00018^)  =  101(1  +  o-oooooS/) 

.  f=      '        L_ 

0-018  —  0-000808   0-017192 

=  58-2°  C. 

3.  In  an  experiment  for  determining  the  absolute  expansion  of  mercury 
by  Regnault's  method,  the  temperature  (T)  of  the  bulb  A  B  is  100°,  that 
of  C  D  and  the  tubes  E  K,  F  L,  is  the  temperature  (/)  of  the  room,  10°  C. ; 
the  height  of  BA  (H)  is  1010  mm.,  DC  (Hj)  is  1009  mm.,  EK  (//)  is 
loi'2  mm.,  and  F  L  (7/t)  is  117*4  mm. :  find  the  dilatation  of  mercury 
between  o°  and  99°  C. 
H 

j  _|_  AT  = (  I  +  A/)   : 

(assuming     r     as  the  mean  dilatation  from  Dulong  and  Petit's  tables). 


,'.  dilatation  from  o°  to  100°  C.  =  AT  =  0*018132 

We  can  now  state  that  0*018132  -i-  -100  is  more  accurate  than    - 

555Q 
for  the  mean  dilatation  ;   we  can  therefore  substitute  this  mean  value. 

/.   i  +  AT  = -(1-0018132)  =  1-018154 

.*.  AT  =  0-018154 
Similarly,  AT  as  a  whole  was  determined  at  the  various  temperatures. 

4.  A  cylindrical  piece  of  glass  (/£  =  0-000008 5)  weighs  4*525  grams  in  air, 
2-817  grams  when  immersed  in  water  at  20°  C.,  and  2'88i  gram?  in  water 
at  100°  C.  :  find  the  dilatation  of  water  between  20°  and  100°  C.  (See 
§350 

^20  =  4'525  -  2*817  =  1708  gram  ;    ^100  =  4-525  -  2-881  =  1-644  gram  ; 
K  =  0-0000255 

.*.  8oK  =  0-00204 

(8cw)  =  7^(J  +  0-00204)  -  i  =  0-041 
.*.  unit  volume  at  20°  C.  =  1-041  at  80°  C. 
(For  further  examples  on  chapter  III.,  see  p.  77.) 


57 


CHAPTER   IV. 
DILATATION  OF  GASES. 

41.  Boyle's  Law.— The  dilatation  of  gases  is  complicated 
by  the  fact  that  not  only  temperature  but  pressure  must  be 
considered.  The  compressibility  of  solids  and  liquids  is  so 
small  that  it  does  not  affect  appreciably  any  of  the  results. 
With  gases  it  is  otherwise. 

The  relation  between  the  volume  and  pressure  of  a  gas  was 
first  formulated  by  Boyle. 

The  experiments  can  be  repeated  by  taking  a  bent  tube 
about  50  inches  long,  closed  at  the  end  of 
the  short  limb,  which  should  be  about  7 
inches.  On  pouring  a  little  mercury  down 
it  can  be  arranged  so  that  it  stands  at  the 
same  level  in  both  limbs.  We  have  then 
a  certain  volume  of  air  cut  off  from  the 
rest  of  the  atmosphere,  and  subject  to  the 
pressure  of  the  atmosphere  ;  this  we  read 
from  the  barometer.  Measure  the  length 
of  the  column  of  air  in  the  short  limb, 
and  read  the  thermometer.  Pour  mercury 
into  the  long  limb.  The  air  in  the  short 
limb  is  compressed.  When  it  cools  to  the 
temperature  of  the  room,  measure  also 
(a)  the  height  of  the  mercury  in  the  long 
limb  above  the  level  of  the  mercury  in  the  short  limb  ;  this, 
with  (b)  the  pressure  of  the  atmosphere,  gives  the  total  pres- 
sure to  which  the  air  is  subjected.  Make  several  such  measure- 


Heat 


ments,  and  record  as  follows, 
millimetres : — 


The  measurements  are  made  in 


V 
Volume. 

(a.)  Pressure 
of  mercury. 

(b)  Pressure  of 
atmosphere 
(barometer). 

P 

Total  pressure 
(*  +  *). 

VXP. 

I 

2 

3 

4 

5 

221 
I76 
112 
102 

95 

H3 

378 
1023 

1200 
1346 

756 

>» 
j) 
If 

» 

899 

H34 
1779 

1956 
2102 

198,679 
199,584 
199,248 

I99.5" 
199,690 

Average 

199,522 

Allowing  for  errors  of  measurements,  we  see  that  the 
general  law  of  Boyle  can  be  accepted — 

If  a  given  mass  of  gas  be  kept  at  the  same  temperature, 
the  product  of  the  volume  and  pressure  is  a  constant ;  or,  the 
volume  varies  inversely  as  the  pressure. 

P  x  V  =  constant 

Clasping  the  short  limb  with  the  warm  hand  produces 
expansion,  and  shows  at  once  the  reason  for  the  insertion  of 
the  words,  "  at  a  given  temperature." 

42.  Isothermals. — These  results  may  be  represented 
graphically  by  taking  OX,  O  Y  as  the  co-ordinates,  measuring 
along  O  X  distances  proportional  to  the  volume,  and  along  O  Y 
distances  proportional  to  the  pressure.  This  is  readily  done 
by  using  squared  paper.  In  Fig.  34  each  side  of  a  square 
represented  a  volume  of  5  mm.  (the  paper  was  divided  into 
-^  inch ;  only  the  inch-lines  are  shown),  and,  measuring 
vertically,  each  side  represented  a  pressure  equal  to  50  mm.  of 
mercury.  Thus  to  represent  the  first  line  of  the  table,  O  a  was 

221 

made   equal  to  =  44*2  sides  of  squares;  a  A  was  made 

equal  to  — -  =  19  sides  of  squares.      Similarly,  the  points 

B,  C,  D,  E  were  obtained  for  lines  2,  3,  4,  and  5  of  the  table. 

F,  G,  H  were  similarly  calculated.      The  points  A,  B,  .  .  . 

G,  H  are  joined  by  a  free  curve  that  passes  through  or  near 


Dilatation  of  Gases 


59 


them  all;  any  point   off  this  curve  probably  denotes  some 
experimental  error. 

If  the  gas  obeys  exactly  Boyle's  law,  the  isothermal  is  a 

Y 


\ 


\ 


\ 


VbLUMES 


75'C 


15' C 

-85X 


O  h  g    f     t     ; 

FIG.  34. 

rectangular  hyperbola.  Practically  the  curves  in  Fig.  34  are 
such  curves. 

This  curve,  representing  the  relation  between  the  volume 
and  pressure  of  a  gas  when  the  temperature  remains  constant, 
is  called  the  isothermal  of  a  given  mass  of  the  substance  for 
that  temperature. 

The  particular  curve  A ...  H  is  an  isothermal  for  air  at  1 5°  C. 

By  surrounding  the  short  limb  of  the  Boyle  tube  with 
a  vessel  containing  water  at  75°,  we  could  similarly  plot  the 
results,  and  obtain  the  isothermal  for  the  same  mass  of  air  at 
75°.  It  would  lie  above  the  curve  drawn,  as  that  for  —85° 
would  lie  below  the  curve. 

If  we  take  any  point,  S,  and  draw  S  s,  then  the  meaning  ot 
the  point  S  is  that  the  pressure  of  the  gas  is  measured  by  S  j, 
and  the  volume  by  O  s  •  in  figures  the  pressure  is  1456  and  the 
volume  137  :  (1456  x  137  =  i99;472)- 


60  Heat 

If  PJ  represent  a  pressure  of  i  atmosphere,  and  Vl  the 
volume  at  that  pressure,  similar  meanings  being  attached  to 
P2  (2  atmospheres),  V2,  etc.,  then  for  gases  that  obey  Boyle's 
law  — 

PiVi  =  PaV2  =  P3V3  =  etc.  =  a  constant 


=  etc'  = 


When  the  pressure  is  between  i  and  2  atmospheres,  it  is 
found  that  at  ordinary  temperatures  gases  such  as  air,  hydrogen, 
nitrogen,  oxygen,  which  we  shall  find  are  gases  that  are  liquefied 

p  v, 
with  difficulty,  practically  fulfil  the  conditions  p-y-  =  i 

In  the  case  of  gases  like  carbonic  acid  and  sulphur  dioxide, 
which  we  shall  find  are  gases  that  are  easily  liquefied  at  ordinary 

PiVj 

temperatures,  the  ratio  p~y~  is  greater  than  unity.  If,  there- 
fore, we  draw  the  isothermal  for  15°  for  sulphur  dioxide, 
beginning  at  a  pressure  of  i  atmosphere,  when  we  reach  a 
pressure  of  2  atmospheres  the  volume  will  be  less  than  that 
given  by  Boyle's  law ;  the  isothermal  will  fall  slightly  below  the 
isothermal  of  a  perfect  gas. 

The  higher  the  temperature,  however,  the  less  the  difference 
becomes.  This  is  readily  seen  from  the  following  values 

p.Vj 

of  the  ratio  ppy.-  for  the  two  gases  at  various  temperatures  : — 

Sulphur  dioxide.  Carbon  dioxide. 

15° 1*0185  5° 1-0065 

50° rouo  50° 1*0036 

ioo°...         ...  1*0054  ioo°...         ...  1*0023 

150° 1*0032  150° 1-0014 

2OO°.. .      ...   l'OO2I       2OO°.. .      ...   I'OOOS 

250°...    ...  1*0016     250°...    ...  1-0006 

It  seems  probable  that  at  yet  higher  temperatures  the 
ratio  will  approximate  to  unity. 

We  can,  then,  use  Boyle's  law  at  ordinary  pressures  for 


Dilatation  of  Gases 


61 


gases  that  are  not  readily  liquefied,  and  for  liquefiable  gases 
for  temperatures  far  above  their  boiling  points. 

43.  Boyle's  Law  at  High  Pressures. — Research  has 
also  been  directed  to  discover  whether  gases  conform  to 
Boyle's  law  at  high  pressures. 

Fig.  35  embodies  Cailletet's  experiments  on  nitrogen  at 
15°  C.  He  found  that  the  value  of  PV,  beginning  at  a  pres- 


U  VOLUMES 

FIG.  35. 

sure  of  52  atmospheres,  was  8184.  It  then  gradually  decreased, 
until,  at  a  pressure  of  78  atmospheres,  it  was  7900;  it  then 
gradually  increased,  and  at  a  pressure  of  252  atmospheres  it  was 
9330.  The  minimum  value  is  indicated  at  M.  A  M  C  is  the 
isothermal  for  nitrogen  at  15°,  B  M  D  the  isothermal  of  a 
perfect  gas  passing  through  M.  The  isothermals  for  oxygen 
and  air  are  similar  to  nitrogen,  and  lie  above  the  isothermal 
of  the  perfect  gas  •  the  minimum  value  of  PV  for  air  is  at  a 
pressure  of  about  83  atmospheres ;  that  of  oxygen  about  132 
atmospheres. 

In  the  case  of  hydrogen  there  is  a  maximum  pressure ; 
its  isothermal  at  high  pressure  would  meet  the  isothermal  of 
a  perfect  gas  at  one  point,  and  would  otherwise  be  entirely 
below  it  (see  also  §  48). 

44.  Gay   Lussae's   Law. — Gay  Lussac  examined    the 


62 


Heat 


dilatation  of  gases  when  the  pressure  was  kept  constant  and  the 
temperature  varied.  He  took  a  thermometer  bulb  and  tube 
carefully  calibrated,  and  estimated,  as  in  §  6,  the 
volume  of  the  tube  and  each  division  of  the  stem. 
Filling  it  with  dry  mercury,  he  inverted  it  over  a 
tube  of  calcium  chloride,  and,  by  moving  a  plati- 
num wire,  made  the  mercury  fall  out  (Fig.  36). 
Its  place  was  taken  by  air  dried  by  passing  over 
the  calcium  chloride.  A  slight  thread  of  mercury 
was  left  in.  This  cut  off  the  imprisoned  air  from 
the  atmosphere,  and  served  as  an  index. 

The  tube  A  B  (Fig.  37)  was  now  placed  in  a 
horizontal  position,  among  melting  ice,  and  the 
position  of  the  index  noted,  the  calcium  chloride 
tube  C  still  being  kept  in  position.  Thus  the 
volume  was  known  of  a  mass  of  air  at  o°  and  at 
the  pressure  of  the  atmosphere.  It  was  now  placed 
in  a  similar  position  in  a  bath  of  water  or  oil  (Fig. 
37),  which  was  gradually  heated,  the  position  at 
various  temperatures  being  observed  from  the  ther- 
mometers E  and  D.  Neglecting  the  slight  increase 
due  to  the  expansion  of  glass,  and  assuming  that  the 
barometer  remained  steady,  Gay  Lussac  was  thus 
able  to  calculate  the  coefficient  of  expansion  of  air  and  other 


FIG.  37. 


gases.    It  was  simply  necessary  to  take  the  volume  at  zero  from 


Dilatation  of  Gases  63 

the  volume  at  /°,  and  divide  by  the  volume  at  zero  ;  this  gave 
the  dilatation  for  /°,  and,  dividing  this  by  /,  he  obtained  the 
mean  for  i°.  He  concluded  — 

(1)  All  dry  gases  have  the  same  coefficient  of  expansion 
if  the  pressure  remains  constant;  and  that  the  coefficient  is 
the  same  at  all  pressures. 

(2)  The  volume   increases   by  a   definite  fraction    of  its 
volume  for  each  rise  in  its  temperature. 

The  first  part  of  the  law  was  discovered  by  Charles,  and  is 
sometimes  known  by  his  name. 

Gay  Lussac  determined  the  coefficient  of  expansion  for 

i°  to  be  0-00365;  this  nearly  equals  —  •  Thus  unit  volume 
at  o°  becomes  i  +  —  at  i°,  i  +  —  at  2°,  and  i  +  -  at 


T 
and  i  +  —  at  T°.    Therefore— 

t 
volume  at  /       *  "*"  J^        273  -f  / 

volume  at  T 

T      -1_ 

T        273  +  T 

273 
If  we  write  the  coefficient  as  a,  then 

V,  =  V0  (i  +  at) 

V,  _   V0(i  +  at}         i  +at        273  +  / 

VT  -  V0(i  +  0T)  ~  i  +  aT  ~  273  +  T 

45.  Absolute  Zero.—  The  law  holds  good  for  some 
distance  below  zero.  Let  us  assume  that  it  holds  good  for 
all  temperatures  below  zero;  then  273  units  of  volume  at  o° 
become  273  —  1,  273  —  10,  etc.,  at  —  1°,  —  10°,  etc.  At  —273° 
evidently  the  volume  would  be  nil.  Now,  we  have  no  know- 
ledge of  such  a  low  temperature,  when  all  the  heat  will  have 
left  the  gas,  and  cannot  conceive  of  a  gas  existing  without 
volume.  But  calling  this  temperature  the  absolute  zero,  and 
measuring  temperatures  from  it,  this  and  other  laws  are  simplified 
in  expression.  o°  C.  becomes  273°;  15°  C,  273  +  15°;  and 
-10°,  263°. 


64  Heat 

We  get  from  §  44 — 

vt  . .  777+T 

/.£,  if  the  pressure  remains  constant,  the  volume  of  a  gas" 
is  proportional  to  the  temperature  measured  from  absolute 
zero. 

The  coefficient  of  expansion,  using  the  Fahrenheit  scale, 

is  nearly  - — ;  therefore  the  absolute  zero  will  be  493  —  32  = 

i"  s  <J 

461  degrees  below  zero  on  that  scale,  that  is,  —461°  F. 

46.  Relation  between  Volume,  Pressure,  and  Tem- 
perature of  Gases. — Suppose  a  given  mass  of  a  gas  to  be 
represented  in  volume,  pressure,  and  temperature  by  Vw  P,, 
and  /.  Now  let  it  change  in  any  way,  the  new  values  being 
V,,  Prt  and  /,. 

V 

V,  at  f  and  P,  pressure  will   be    i     *fff  at   o°   and   P, 

pressure. 

V, 

V,  at  t°  and  P,  will  be  j     *af   at  o°  and  P,  pressure. 

Both  are  at  the  same  temperature,.  o°  C. 

V  V,, 

Therefore  by  Boyle's  law,     _±_      .  P,  =  .     ,        .  P,,  =  for 

any  other  temperature  4°,  and  pressure  Pr,  -      -7.  Pr;  thus 

I    ~~|~  Clif) 

*  *     for  any  given  mass  of  gas  is  a  constant. 
This  combines  the  two  laws.     If  we  measure  from  absolute 
zero,   we    have  -         —  =  constant  X  273  =  constant,     and 

VP 

-TJFT    becomes   a  constant,    if  T   represents   the   temperature 

measured  from  absolute  zero. 

,'.  relation  between  pressure  and  temperature  of  gases  is — 

V  P         """  V  P  VP 

v  oro  v  t*-t  v  * 

=  constant 


+  a.o        \+at        T 


Dilatation  of  Gases 
If  the  volume  does  not  vary,  we  have—^ 


T  -f  at 


a  - 


a  is   the  coefficient   of   the  increase   of   pressure   when 
volume  remains  constant. 

If  the  pressure  remains  constant,  then— 


V   --^- 
V°  ~   i  +  at 


a  - 


a  is  the  coefficient  of  the  expansion  of  gases  (increase  of 
volume)  under  constant  pressure.  It  is  the  coefficient  already 
found  by  Gay  Lussac. 

If  Boyle's  and  Charles's 
laws  were  rigorously  true,  the 
two  coefficients  would  be 
equal.  A  further  examination 
is  due  to  Regnault 

47.  Relation  between 
Pressure,  Temperature, 
and  Volume. — Regnault's 
Method.  —  (A)  Constant 
Volume.— (i)  The  capacity  of 
a  bulb,  A  (Fig.  38),  and  stem 
to  a  certain  mark,  a,  are  ac- 
curately determined  at  o°  and 
1 00°.  The  stem  is  calibrated, 
and  therefore  the  volume  of 
the  distance  a  b  is  known. 

(2)  The  bulb  and  stem  are 
filled  with  dry  gas. 

(3)  The  coefficient  of  ex- 
pansion  of  the  glass  is  known. 

The  stem  is  fixed  into  a 
wider  vertical  tube,  M,  which,  in  turn,  is  connected  with  a  similar 
vertical  tube,  N,  open  at  the  top.  Mercury  can  be  poured  into 
the  top  of  N  or  be  taken  from  the  apparatus  by  turning  the 
tap  R,  when  it  flows  into  the  vessel  D. 


FIG.  38. 


66  Heat 

The  bulb  and  stem  to  a  are  first  surrounded  with  melting 
ice.  By  pouring  mercury  into  N,  or  by  running  it  off  at  R,  it  is 
arranged  that  the  mercury  rises  to  b.  Suppose  its  position  in 
the  other  tube  is  c.  There  is  then  a  mass  of  imprisoned  air, 
whose  volume  is  V0  (the  bulb  and  stem  to  a)  at  a  temperature 
of  o°  C.,  and  v  (the  stem  from  a  to  b)  at  a  temperature  of  t°, 
that  of  the  room.  This  gas  is  subjected  to  a  pressure  of  the 
atmosphere  -f-  the  pressure  due  to  a  column  of  mercury  whose 
height  is  the  difference  of  the  levels  of  c  and  b.  Let  the  sum 
of  these  pressures  be  P.  The  temperature  of  the  room  will 
be  slightly  above  zero,  so  that  the  gas  in  the  tube  is  subject  to 

p 
a  pressure  equivalent  to  —  _,    *     at  o°,  where  A  is  the  coeffi- 

cient of  expansion  of  mercury.     Call  this  P0. 

The  bulb  and  stem  to  a  are  now  surrounded  with  a  steam 
bath  whose  temperature  is  T°,  differing  but  slightly  from  100°. 
The  air  in  the  bulb  expands  ;  but  by  pouring  mercury  into  N,  the 
mercury  is  forced  back  to  the  mark  £,  rising,  of  course,  higher 
in  N  to  some  position,  ^.  The  gas  is  now  subject  to  a  pres- 
sure, Q,  which  equals  the  barometric  height  -f  the  difference  in 

levels  of  b  and  ^.    This  reduced  to  o°  =  Y+  A?     Cal1  this  Qo 

The  volume  of  the  bulb  and  stem  to  a  is  now  V0(i  -f  KT), 
K  being  the  coefficient  of  cubical  expansion  of  glass.  From 
a  to  b  the  volume  is  v\  this  we  can  assume  remains  constant 
during  the  experiment. 

In  the  first  part  we  have  a  volume  of  air,  V0  +  —  ^  —  , 

pressure  P0,  and  temperature  o°  C.  ;  in  the  second  part,  reduc- 
ing all  to  o°  C.,  there  will  be  a  volume— 

V0(i  +  KT)  v 

!+aT~  +  r+^Sf  at  °°  and  Pressure  Q° 

a  is  the  mean  coefficient  of  expansion  of  dry  air  for  i° 
Applying  Boyle's  few  — 

/Vo(i+KT)  v 


a  is  the  only  unknown  quantity. 


Dilatation  of  Gases  67 

Regnault  by  this  method  found  a,  the  coefficient  of  pressure 

at  constant  volume,  to  be  0*003663.     Its  value  in  the  previous 

p  __  p 
section  is  given  as  a  =     *  p     ° 

(B)  Constant  Pressure. — The  experiments  were  repeated, 
arranging  so  that  the  pressure  should  remain  constant.  This 
was  attained  by  seeing  that  the  mercury  in  the  experiments 
stood  at  the  same  level  in  each  tube ;  the  gas  was  thus  sub- 
jected to  the  pressure  of  the  atmosphere.  The  gas,  in  ex- 
panding, was  forced  down  M,  and  this  being  large  compared 
with  the  bulb,  it  was  necessary  to  know  its  exact  temperature ; 
M  and  N  were  therefore  surrounded  by  a  bath  of  water  whose 
temperature  was  accurately  known. 

The  calculation  was  similar  to  the  former,  and  Regnault 
found  the  coefficient  of  expansion  at  constant  pressure  to  be 

y   Y 

a  =     \7  .   °;  but  the  value  obtained  was  ©'00367  (cf.  §  46). 

V  (J 

The  following  results  were  obtained  : — 


Hydrogen 
Air 
Nitrogen 
Carbonic  acid 

Coefficient  of 
increase  of  pressure 
at  constant  volume. 

0*003667 
...       0*003665 
...       0*003668 
...       0-003688 

Coefficient  of 
increase  of  volume 
at  constant  pressure. 

©'003661 
0*003670 
0*003670 
...          0*0037IO 

Sulphurous  acid...     0*003845       ...       0*003903 

These  numbers  were  calculated  from  the  dilatation  between 
o°  and  100°  C. 

Further  researches  showed  that  the  numbers  increased  wilh 
the  pressure ;  for  example,  the  coefficient  of  increase  of  volume 
of  air  under  constant  pressure,  at  a  pressure  of  7  60  mm.  was 
0*00367  ;  at  a  pressure  of  2620  mm.  it  became  0*003696. 

Sulphurous  acid  rose  from  0*00390  at  760  mm.  pressure 
to  0*00398  at  980  mm.  pressure. 

Air  was  examined  carefully  at  constant  volume,  beginning 
with  various  pressures.  A  given  mass  of  dry  air  was  subjected 
to  a  certain  pressure,  measured  in  terms  of  the  height  of  the 
mercury  column  above  or  below  the  pressure  of  the  atmo- 


68  Heat 

sphere  at  o°  C. ;  it  was  then  heated  to  100°,  and  the  pressure 
was  read  after  seeing  that  the  gas  occupied  the  original  volume. 
The  coefficient  of  the  increase  of  pressure  was  then  found 
after  necessary  calculations  and  corrections. 

Pressure  at  o°.  Pressure  at  100°.  Coefficient. 

109*72  I49'3I  •••  0*003648 

266*06  ...       395'O7  •••  0*003654 

760*00  ...      1038-54  ...  0-003665 

2144-18  ...      2924-04  ...  0*003689 

The  changes  in  the  case  of  gases  that  are  easily  liquefied 
will  be  discussed  later. 

48.  Deviations  from  the  Two  Gaseous  Laws.— 
(Read  §§  42  and  43.)      The  following   results,  taken    from 
Amagat's  researches,  will  show  the  increased  resistance  gases 
offer  to  compression  at  high  pressure.     The  temperature  was 
at  15°  C.  throughout,  and  in  each  case  the  gas  occupied  unit 
volume   at   a  pressure  of   i  atmosphere.      The   last  column 
gives  the  volume  a  gas  would  occupy  at  each  pressure,  if  it 
obeyed  Boyle's  law  : — 

Pressure  in  Volumes, 

atmospheres.  Air.  Nitrogen.  Hydrogen.  Perfect  gas. 

75O      O"OO22O      O-OO262  •  0*OOI33 

1000    0*00197    0*00203    0*00169    0*00100 

I5OO      O*OOI7I      O*OOI76      O*OOI34     0*00067 

2000    0*00157    0*00161    0*00116    0*00050 

25OO      0-OOI47      O*OOI5I      O*OOI05      O*OOO4O 
3OOO      O*OOI4O      O-OOI45      0*00096      0*OOO3O 

The  deviations  from  Charles's  law  are  more  marked.  This 
is  especially  the  case  at  high  temperatures,  when  an  increase 
in  temperature  is  followed  by  a  greater  increase  in  volume 
than  the  simple  form  of  the  law  accounts  for. 

49.  Joly's  Air-Thermometer. — The  coefficient  of  gases 
being  much  higher  than  that  of  liquids,  and  being  moreover 
fairly  constant  between  certain  limits,  has  suggested  the  use  of 
gases  in  thermometry.     A  simple  form  is  shown  in  Fig.  85  ; 
the  pressure  will,  however,  vary  for  every  position  of  the  fluid, 
and  this  leads  to  calculations  for  each  reading. 


Dilatation  of  Gases 


The  air-thermometer  of  Joly  (Fig.  39)  is  an  adaptation  of 
Regnault's  apparatus. 

a  is  a  glass  bulb  with  a  fine  tube  bent  twice ;  its  volume 
is  V  (about  50  c.c.  capacity).  This  is  joined  to  another 
tube,  R,  supported  by  a  block,  A.  R  is 
connected  by  indiarubber  tubing  to  a  longer 
open  tube,  R',  supported  by  a  block,  A'. 
R,  R'  and  the  indiarubber  tube  contain 
mercury,  a  is  filled  with  dry  air  and  is 
surrounded  with  melting  ice.  R'  is  raised 
or  lowered  until  the  mercury  rises  to  a  fixed 
mark,  S.  Then  we  have  a  volume  V,  at 
temperature  o°  C.  and  pressure  P,  equal  to 
that  of  the  atmosphere,  together  with  (or  less 
than)  that  due  to  the  column  of  mercury 
whose  vertical  height  is  the  distance  between 
S  and  the  level  of  the  mercury  in  R'.  Then, 
neglecting  the  small  portion  of  the  tube  not 

VP 

surrounded  with  melting  ice,  -™r  =  a  con- 
stant, C,  by  Boyle's  and  Charles's  laws.  This 
first  measurement  gives  C  when  necessary. 

If  now  the  bulb  be  placed  in  an  enclosure 
whose  temperature  is  /°,  the  column  of  the 
glass  becomes  V(i  -f-  K/),  where  K  is  the  coefficient  of  the 
cubical  expansion  of  glass.     The  total  pressure,  /,  is  noted 
as  before,  after   the   mercury  is   brought   to   the  level  S  by 
raising  or  lowering  R'.     If  a  be  the  coefficient  of  expansion  of 
air,  then  — 

V(i  +  K/)/  _  VP 
i  +  at         T 


FIG.  39- 


This  is  the  same  formula  as  in  §  47,  omitting  slight  cor- 
rections, and  Joly,  by  his  apparatus,  determined  the  coefficient 
of  expansion,  a.  Also — 

._    /-P 

aP  -  K/ 


7O  Heat 

Substituting  the  accepted  values  of  a  and  K,  and  deter- 
mining P  for  the  particular  thermometer,  then  — 


0*003665?  —  0*000025^ 

i.e.  the  temperature  is  determined  from  the  pressure  p. 

50.  Density  of  Gases.  —  The  density  of  a  substance  is 
the  mass  of  unit  volume. 

If  V  be  the  number  of  units  of  volume  in  a  substance,  D 
the  density,  then  — 

VD  =  mass  =  m 

If  the  volume  changes  to  Vj,  and  the  density  to  D1?  then  — 
VD  =  ;;/  =  VjA  /.  ^  =  § 

i.e.  the  density  is  inversely  as  the  volume.  Also,  if  T  be  the 
temperature  measured  from  absolute  zero  — 

X?  _  ViPi  V         D!        PtT 

T    :      Tx  '"•  Vx  ~  D  ~  PTj 

P  PI  P2 

.'.  y^f,  =  -pv  rp   =  -p.  rp~  =  etc.  =  constant 

JJ  1  I/!  1  1  L/2  1  2 

p 
and  D  =  7p  X  constant 

or  density  varies  directly  as  the  pressure  and  inversely  as  the 
absolute  temperature.  Therefore,  in  dealing  with  densities, 
the  pressure  and  temperature  must  be  considered.  .  This  is 
especially  the  case  in  gases. 

In  liquids,  the  temperature  must  be  allowed  for  ;  the  effect 
of  pressure,  however,  is  small  and  can  generally  be  neglected. 
In  solids,  while  temperature  must  be  considered  in  accurate 
calculations,  the  effect  of  pressure  can  generally  be  neglected. 

51.  Density  of  Gases.—  Regnault's  Method.  —  The 
specific  gravity  of  the  gas  is  determined  relative  to  air. 

Two  globes  made  of  the  same  kind  of  glass,  as  nearly 
equal  in  volume  as  possible,  and  each  of  about  10  litres 
capacity,  were  used. 

(i)  Experiments  were  made  to  ensure  that  the  two  globes 


Dilatation  q}  Gases  71 

were  of  equal  volume,  so  that  on  hanging  them  from  the  two 
arms  of  a  balance,  equal  volumes  of  air  were  displaced,  and 
the  corrections  for  weighing  in  air  were  unnecessary. 

(2)  One  globe  was  placed  in  melting  ice  (Fig.  40).  A 
three-way  stop-cock,  A,  enabled  the  globe  to  be  connected 
either  with  an  air-pump  or  with  drying-tubes,  M  N,  through 
which  the  gas  entered  by  C.  The  following  operations  are 


FIG.  40. 

performed :  the  air  is  exhausted  through  D  ;  then  A  is  turned 
so  that  dry  gas  enters  through  M  N  while  D  is  shut  off.  The 
drying-tubes  are  shut  off,  and  the  air  and  gas  are  again  drawn 
off;  this  is  repeated  until  the  globe  contains  dry  gas  only, 
without  any  admixture  of  air.  The  gas  is  now  exhausted  as 
far  as  possible,  until  the  pressure  (measured  by  a  differential 
barometer  connected  by  E)  becomes  small :  let  this  pressure 
be/. 

B  is  closed ;  the  globe  is  removed,  cleaned,  and  weighed, 
with  the  other  globe  as  a  counterpoise.  Shot  is  placed  in 
the  scale  of  B  until  there  is  a  perfect  balance.  The  globe 
contains  a  mass  of  gas  whose  volume  is  V,  at  temperature 
o°  C.,  and  pressure/  millimetres  of  mercury. 


72  Heat 

The  globe  is  now  replaced  in  melting  ice.  The  stop-cock 
is  turned  so  that  dry  gas  enters  through  M  and  N  ;  the  stop-cock 
is  kept  open  until  the  gas  is  at  the  pressure  of  the  atmosphere, 
P,  read  from  the  barometer.  B  is  closed,  and,  in  a  similar 
manner  to  the  above,  a  perfect  balance  is  made  by  adding  a 
weight,  m,  to  the  other  scale.  m  is  the  mass  of  the  gas  that 
has  entered  during  the  second  operation,  that  is,  it  is  the  mass 
of  a  volume,  V,  at  temperature  o°  C.,  and  pressure  (P—  /)  milli- 
metres of  mercury.  An  equal  volume  of  the  gas  at  standard 
pressure  would  weigh  — 


Exactly  similar  operations  are  repeated  for  dry  air,  and  a 
weight,  m',  is  obtained  which  is  the  mass  of  a  volume  of  dry 
air  (V)  at  temperature  o°  C.  and  under  a  pressure  (F  —  /). 
The  mass  of  this  volume  at  standard  pressure  would  be  — 


The  specific  gravity  of  the  gas,  compared  with  air  under 
similar  conditions,  is  — 

760 


,       760 
m  .  =f 


The  following  relative  densities  of  gases  have  been  ob- 
tained by  this  method  : — 

Air i -ooooo  Hydrogen          ...  0-06926 

Oxygen       ...     i '10563  Carbonic  acid  ...   1-52901 

Nitrogen     ...     0-97137 

If  the  density  of  air  were  known  at  the  normal  temperature 
and  pressure,  the  density  of  any  gas  whose  specific  gravity  is 
known  could  be  at  once  calculated.  In  the  above  experiments 
the  weight  of  a  certain  volume,  V,  of  air  at  o°  C.  and  760  mm. 
pressure  has  been  found  to  be — 


Dilatation  of  Gases  73 

The  density,  D  =  ±- .    It  is  therefore  necessary  to  determine 

the  exact  volume  of  the  globe  under  similar  conditions  of 
temperature  and  pressure. 

(1)  The  globe  is  filled  (observing  the  former  precautions) 
with  dry  air  at  a  temperature  o°  C.  and  pressure  P.   It  is  weighed 
without  using  the  second  globe  as  a  counterpoise.    The  weight 
m  gives  the  mass  of  the  globe  and  the  air  under  the  above 
conditions. 

(2)  The  globe  is  filled  with  water,  a  method  analogous  to 
filling  a  thermometer    being  followed.      A  curved    tube    is 
fitted  to  the  globe,  one  end  dipping  under  boiling  water ;  on 
heating  the  globe  to  expel  some  of  the  air,*and  allowing  the 
globe  to  cool,  part  of  the  water  is  forced  over  into  the  globe ; 
this  is  boiled,  the  air  is  chased  out,  and  on  cooling  the  boiling 
water  siphons  over.      The  globe  after  cooling  is  placed  in 
melting  ice  for  twenty-four  hours ;  the  stop-cock  is  closed,  the 
globe  cleaned  and  left  for  several  hours  in  a  room  whose  tem- 
perature is  below  9°.     It  is  weighed  in  this  room  again  without 
the  balancing  globe.  If  the  weight  is  m',  the  barometer  being  P' — 

m1  —  m  =  (weight  of  globe  +  water)  —  (weight  of  globe  +  air) 
=  weight  of  water  —  weight  of  air 

i.e.  the  weight  of  water,  W  =  m'  —  m  +  weight  of  air. 

The  weight  of  air  is  M  (see  above),  it  being  found  that  the 
correction  for  the  expansion  of  glass  and  air,  and  for  the  air 
displaced  in  weighing,  could  be  neglected. 

Thus  the  weight  of  water  at  o°  which  the  globe  at  o°  would 
contain  is  known.  Unit  volume  of  water  at  4°  =  1*000017  at 

o°  C.,  i.e.  i  cubic  centimetre  of  water  at  o°  weighs 

1-000017 

gram  ;  or  i '000017  c-c-  at  °°  weighs  i  gram.  Therefore  W,  the 
weight  of  water  obtained  above  in  grams,  represents  a  volume, 
W  x  i  'oooo  1 7  c.c.  Call  this  V,  a  quantity  which  now  is  known. 

(a)  The  mass  of  the  air  is  known  (M). 

(b)  The  volume  of  this  air  is  known  (V). 

.'.  the  density  of  dry  air  =  the  mass  of  i  c.c.  =  M  4-  V  can 

be  calculated 


74  Heat 

This  was  found  to  be  o '0012 93 187  gram. 

i  litre  (1000  c.c.)  of  dry  air  at  o°  C.  and  pressure  of  760 
mm.  of  mercury,  weighs  1*2932  gram. 

In  the  following  table  the  first  column  gives  the  absolute 
density  of  the  gas,  that  is,  the  mass  of  i  c.c.  at  o°  C.  and  at  760 
mm.  pressure;  the  second  column  gives  the  relative  density 
compared  with  air  under  similar  conditions;  the  third  the 
relative  density  compared  with  hydrogen  : — 


Density. 

Air  =  i. 

Hydrogen  = 

Air 

0*0012932 

...        1*0 

...        I4'438 

Oxygen  .  .  . 

0*0014298 

I'IO56 

...        15-96 

Hydrogen 

0*0000896 

...        0*0693 

1*0 

Nitrogen 

O*OOI256l 

...        0-9714 

I4'O 

Carbonic  acid 

0*0019774 

...         1-5290 

22'0 

Marsh-gas 

0*0072700 

...        0-5590 

...          8'0 

Ammonia 

0-0076970 

0'59IO 

...       8-5 

52.  Boyle's  and  Charles's  Laws.  —  Special  Case.  — 

In  the  case  of  air,  if  the  units  be  i  foot  and  i  lb.,  and  the  tem- 
perature scale  that  of  Fahrenheit,  we  have  the  following 
experimental  results  :  — 

i  lb.  of  air  at  32°  F.  and  at  normal  atmospheric  pressure 
of  14*7  Ibs.  per  square  inch,  or  2116-8  Ibs.  per  square  foot, 
measures  12*387  cubic  feet. 

PV  2116-8  x  12*387       P^ 

.'.  -~r  =  constant  =  -  —/r~L  =  ~^r~ 

492-6  1\ 


. 

..    ^ 

.*.  at  any  absolute  temperature  Tl  — 


If,  therefore,  any  of  the  two  quantities  P,  V,  T,  be  known, 
the  third  can  be  readily  calculated. 

If  a  number  of  calculations  are  needed  for  hydrogen  or 
other  gas,  we  again  determine  P  and  V  at  some  known  tern- 
perature. 

1000   c.cm.    of  hydrogen   at   o°    C.    and    760   mm.    pressure 
=  0-089578  gram 


Dilatation  of  Gases  75 

If  the  units  be  i  centimetre,  i  gram,  and  1°  C— 


1000 

=  «  '63-45  c.c. 


0-089578 

A  column  of  mercury  760  mm.  high  will  exert  a  pressure 
of  1033-296  grams  per  square  centimetre. 

^o  =  losr^ii^s  =  422i7 

.*.  pV   =   4221'jT 

53.  Gases  at  Normal  Temperature,  T0  (o°  C.),  and 
Pressure,  P0  (760  mm.). — The  necessary  instructions  for 
changing  the  volumes  of  gases  at  any  temperature  and  pressure 
to  the  normal  temperature  and  pressure  have  already  been 
given  (§  46).  It  is,  however,  necessary,  on  account  of  the 
importance  of  the  change,  to  again  examine  the  process. 

Let  the  volume  be  Vt  in  c.cm.,  the  pressure  P,  in  milli- 
metres of  mercury,  and  the  temperature  f  C. 

(1)  Let  the  pressure  remain  constant  at  Pt,  while  the  tem- 

y 
perature   changes  to   o°,   then   the   new  volume  V  =         ' 

by  Charles's  law. 

(2)  Let  the  pressure  change  from  Pt  to  P0,  the  temperature 
remaining  at  o°  C.     If  V0  be  the  new  volume — 

VP  —  v  P 

*  •*•  t  ~     *  0    0* 

PV      P 

•  v       v   —  — 

••     °  ~      *P0  "  (i  -f  «/)P0 

a  -  coefficient  of  expansion  -  0-003665  for  all  gases;  P0  =  760 

v  .  P 
v 

*  *     °  ~  (i  4-  0-003665/;  •  760 

.WORKED  EXAMPLES. 

I.  The  volume  for  a  certain  quantity  of  air  at  50°  C.  is  500  cubic 
inches.  Assuming  no  change  of  pressure  to  take  place,  determine  its 
volume  at  —50°  C.  and  at  100°  C.  respectively. 

—  constant  =  ~~- 
'.'  pressure  is  constant,  .*.  /t  =  /, 


76  Heat 

To  273  —  50      500  x  223 

.-.  volume  at  -50'  C  .  «-  t        =  500  •  =  - 


=  345*2  cubic  inches 
Volume  at  100°  C.  =  v2  =  500  -  ^  =  577-4  cubic  inches 

Or,  since  pressure  is  constant  by  Charles's  law,  the  volumes  vary  as  the 
absolute  temperature. 

volume  at  —50°  _  273  -  50  _  223 
*  '     volume  at  50°    ~  273  +  50  ~  323 

/.  volume  at  -  50°  =  volume  at  50°  X  ^ 

=  5oox^ 
323 
=  345*2  cubic  inches 

2.  A  quantity  of  gas  collected  in  a  graduated  tube  over  mercury 
measures  305-6  c.cm.  j  the  mercury  in  the  tube  is  47  mm.  above  the  level 
in  the  dish;  the  temperature  of  the  room  is  15°  C.,  and  the  barometer 
reads  762  mm.  :  find  the  volume  of  gas  at  o°  C.  and  760  mm.  pressure. 

The  305-6  c.cm.  is  subjected  to  a  pressure  of  762  —  47  =  715  mm.  of 
mercury. 

(1)  Its  volume  when  the  pressure  is  760  mm.  =  305-6  X  '-g-  at  15° 

(2)  ,,  ,,  ,,  760  mm.  and  the  temperature  is  o°  C. 

305-6  X  715 

760  X  (i  +  0-003665  X  15) 
Or,  by  using  the  formula  in  §  53  — 


=  etc. 


Volume  =  =  as  before 


3.  A  glass  flask  was  filled  with  dry  air  when  the  temperature  of  the 
room  was  10°  C.  and  the  barometer  read  750  mm.  The  stop-cock  was 
closed  and  the  flask  weighed.  It  was  now  placed  in  a  bath  of  water  at 
100°  C.  The  stop-cock  was  opened,  afterwards  closed,  and  when  the  flask 
cooled  down  it  was  again  weighed.  It  was  found  that  2  grams  of  air  had 
escaped.  Find  the  capacity  of  the  flask  at  10°  C.  and  at  o°  C.  (coefficient 
of  cubical  expansion  of  glass  =  0*000025). 

Let  z>0,  z>10,  vloo  =  volume  of  flask  at  o°,  io°,.and  100°  C.  in  c.cm. 
v\9  =  v0(i  +  0*00025)  =  volume  of  air  at  10° 
z'loo  =  »0(I  +  0-0025) 


Dilatation  of  Gases 


the  air  that  escapes  =  ?0(i  '00025         _  1-0025) 


Dividing  this  by  the  density  of  air  at  100°,  that  is,  0*001293  *  ^^ 
we  get  the  mass  of  this  air.     This  equals  2  grams. 

89*3857  x  373 
•'•  2 


•    283  x  273  x  0-001293 

2  X  283  X  273  X  0-001293 

••"0=  89-3857  x  373  --  =  Ans.inc.cm. 

z>JO  equals  this  quantity  multiplied  by  I  '00025 


EXAMPLES.     III. 

1.  Explain  clearly  the  difference  between  real  and  apparent  expansion 
of  a  liquid.     What  quantity  do  you  neglect  in  stating  that  A  =  D  +  G  ? 

If  the  coefficient  of  cubical  expansion  of  the  solid  were  0-003,  and 
the  coefficient  of  apparent  expansion  of  the  liquid  were  0*04,  would  the 
above  formula  be  true?  Find,  under  these  conditions,  the  coefficient  of 
absolute  expansion  of  the  liquid. 

2.  A  glass  flask  contains  when  full  at  O°  C.   lOO  c.cm.  of  mercury. 
The  coefficient  of  cubical  expansion  of  glass  being  0-000026,  and  that  of 
mercury  o'oooiS,  find  the  volume  at  100°  C.  of  the  mercury  driven  out 
when  the  flask  and  mercury  are  heated  to  100°.     (London  Matric.,  1885.) 

3.  The  barometer  is  read  as  764/5  mm.  in  a  laboratory  when  the 
temperature  is  12°  C.     State,  in  order  of  their  importance,  the  corrections 
that  should  be  made. 

4.  How  would  you  determine  the  coefficient  of  apparent  expansion  and 
of  absolute  expansion  of  glycerine  ? 

5.  Water  at  60°  F.  is  exposed  to  air  at  30°  F.  ;  it  gradually  cools  until 
it  reaches  the  temperature  of  the  air :   describe  fully  the  changes  which 
occur  in  the  water  during  the  whole  process  of  cooling. 

6.  Describe  Dulong  and  Petit's  method  of  determining  the  absolute 
expansion  of  mercury. 

In  an  experiment  one  tube  was  at  o°  C.,  the  other  was  at  150° ;  the 
heights  of  the  columns  of  mercury  were  36-00  inches  and  36-97  inches : 
calculate  the  coefficient  of  absolute  expansion  of  mercury. 

7.  State  exactly  in  what  respects  Regnault's  method  was  an  improve- 
ment upon  Dulong  and  Petit's  method. 

8.  State  clearly  the  meaning  of  "mean  coefficient  of  expansion  "  and 
"  true  coefficient  of  expansion." 

9.  Represent  graphically  the  results  of  the  tables  on  pp.  45  and  50 


78  Heat 

EXAMPLES.     IV. 

1.  What  relation  exists  between  the  temperature,  pressure,  and  volume 
of  a  given  quantity  of  gas  ?    A  cubic  foot  of  air  at  100°  C.  is  cooled  down 
to  o°,  and  at   the  same  time  its  pressure  is  halved  :   determine  its  new 

volume,     (a  —  — j 

2.  Describe  Regnault's  experiments  and  apparatus  to  determine  the 
coefficient  of  expansion  of  a  mass  of  air  under  constant  pressure  ;  and  state 
the  conclusion  that  he  drew  from  the  fact  that  it  differs  from  the  coefficient 
of  increase  of  elasticity  under  constant  volume. 

3.  If  a  litre  of  air  at  o°  C.  and  a  pressure  of  76  cm.  of  mercury  has  a 
mass  of  1*293  gram>  determine  the  mass  of  a  cubic  metre  of  air  measured 
at  50°  C.  and  a  pressure  of  50  cm. 

4.  Describe  a  method  of  finding  the  coefficient  of  increase  of  elasticity 
of  a  mass  of  air  occupying  a  constant  volume. 

5.  A  solid  at  o°  C.,  when  immersed  in  water,  displaces  500  cubic  inches  ; 
at  30°  C.  it  displaces  503  cubic  inches :  find  its  mean  coefficient  of  linear 
expansion.     (London  Matric.,  1880.) 

6.  A  drying-oven  measures  internally  20  cm.  X  18  cm.  X  15  cm.  ;  the 
temperature  is  120°  C. ;  the  barometer  reads  750  :  find  the  mass  of  air  con- 
tained in  the  oven. 

7.  10  cubic  feet  of  dry  air  measured  at  32°  F.  when  the  barometer  is 
29  inches  are  forced  into  a  globe  whose  diameter  is  10  inches ;  the  globe  is 
heated  to  the  temperature  of  steam  at  normal  pressure  :  find  the  pressure 
of  the  globe. 

8.  10  grams  of  nitrogen  are  placed  in 'a  vessel  whose  capacity  is  1000 
litres,  the  temperature  being  17°  C.  :  find  the  pressure  on  the  vessel. 

9.  What  is  an  isothermal  ?    Apply  your  answer  to  graphically  repre- 
senting Boyle's  law. 

10.  State  any  deviations  you  know  from  Boyle's  law. 

11.  Find  the  absolute  density  of  dry  air  at  o°  C.  and  76  cm.  pressure 
from  the  following  data,    obtained  by    Regnault   by  his   compensation 
method  : — 

When  the  globe  was  filled  with  dry  air  at  o°  C.  and  76-199  cm. 
pressure,  and  1-487  gram  added,  they  equilibrated  the  compensation 
globe  and  the  counterpoise.  After  being  placed  in  melting  ice,  and 
exhausted  till  the  pressure  of  the  residual  air  was  0*843  cm->  the  globe 
and  its  contents,  together  with  14-151  grams,  equilibrated  the  compensation 
globe  and  its  counterpoise.  At  6°  C.  and  76*177  cm.  pressure  the  globe, 
when  open,  weighed  1258*55  grams. 

When  it  was  filled  with  water  at  o°  C.,  and  weighed  at  6°  C.  and 
76*177  cm.  pressure,  it  weighed  11,126-06  grams.  (Day's  Examples.) 


79 


CHAPTER   V. 
HEAT  AS  A    QUANTITY. 

54.  Unit  of  Heat. — In  the  previous  chapter  the  change 
in  temperature  alone  has  been  observed  Heat  has  been  given 
to  or  taken  away  from  substances,  but  there  has  been  no 
attempt  made  to  measure  the  amount. 

A  certain  quantity  of  heat  is  required  to  raise  the  tempera- 
ture of  i  Ib.  of  water  from  o°  to  i°  C.  ;  under  the  same  con- 
ditions this  quantity  of  heat  will  always  raise  i  Ib.  of  water 
from  o°  to  i°  C.  ;  if  the  temperature  under  the  same  conditions 
fall  from  i°  to  o°  C.,  then  an  equal  quantity  of  heat  will  be 
liberated.  If  we  experiment  with  2  Ibs.  of  water,  we  shall  deal 
with  twice  the  amount  of  heat;  or,  with  10  Ibs.  of  water,  ten 
times  the  original  amount  of  heat. 

Assuming  for  the  present,  what  is  nearly  true,  that  the  same 
amount  of  heat  is  required  to  raise  the  temperature  of  i  Ib.  of 
water  through  i°,  beginning  at  o°,  i°,  2°,  10°,  60°,  etc.,  we 
can  state  that  it  would  take  six  times  the  amount  of  heat  to 
raise  i  Ib.  of  water  from  o°  to  6°  that  it  takes  to  raise  i  Ib. 
from  o°  to  i° ;  and  fifty  times  the  amount  to  raise  5  Ibs.  from 
50°  to  60°  that  it  takes  to  raise  i  Ib.  from  o°  to  1°. 

It  is  also  assumed  that  the  amount  of  heat  necessary  to 
raise  a  given  mass  of  a  substance  through  a  given  range  of 
temperature  equals  the  amount  of  heat  liberated  when  the 
given  mass  cools  from  the  higher  to  the  lower  temperature. 

Heat,  then,  is  a  measurable  quantity.  It  is  necessary  to 
decide  upon  a  unit,  the  particular  unit  selected  being  a  matter 
of  convenience. 

In  England  the  unit  of  heat,  or  thermal  unit,  is  the  amount 


80  Heat     • 

of  heat  required  to  raise  i  Ib.  of  water  under  the  pressure  of 
one  atmosphere  from  50°  to  51°  Fahrenheit. 

For  practical  purposes  the  effect  of  pressure  on  water  can 
be  neglected,  and  the  specification  of  the  particular  tempera- 
ture is  unnecessary,  and  the  unit  of  heat  becomes  the  quantity 
of  heat  required  to  raise  the  temperature  of  water  through  one 
degree.  The  other  particulars  should,  however,  always  be 
understood. 

For  scientific  purposes  the  unit  of  heat  is  the  amount  of 
heat  required  to  raise  i  gram  of  water  under  a  pressure  of  one 
atmosphere  from  o°  to  i°  Centigrade. 

In  the  following  calculations  it  will  be  assumed  that  the 
unit  of  heat  is  the  amount  required  to  raise  i  gram  of  water 
through  i°  Centigrade  ;  that  is,  that  the  small  variation  due  to 
change  of  atmospheric  pressure  and  due  to  temperature  will 
be  neglected. 

If  5  Ibs.  of  water  at  o°  be  mixed  with  5  Ibs.  of  water  at  100°, 
the  result  is  10  Ibs.  at  50°  ;  the  former  has  taken  up  250  units, 
the  latter  has  given  up  250  units.  10  Ibs.  of  water  at  o°  +  20  Ibs. 
at  30°  =  30  Ibs.  at  20°  ;  the  former  has  taken  (10  X  20)  units 
of  heat  that  have  been  disengaged  from  the  20  Ibs.  cooling  10° 
and  giving  up  200  units. 

Generally,  if  a  mass  m±  of  water  at  t°  be  mixed  with  a  mass 
mz°  at  /2°  (a  higher  temperature),  the  resulting  temperature  will 
be  0  ;  so  that— 

mi(0  -  A)  *  ™2(4  -  0) 
.         mJ*  +  ffliA 


55.  Specific  Heat.—  On  mixing  i  Ib.  of  turpentine  at  say 
15°  C.  with  i  Ib.  of  water  at  100°  C.,  the  result  is  a  mixture  of 
2  Ibs.  at  about  75°  C.  ;  i.e.  i  Ib.  of  turpentine  has  been  heated 
through  60°  C.  by  25  units  of  heat. 

To  raise  i  Ib.  of  turpentine  i°  requires,  then,  0*42  unit  of 
heat  nearly. 

The  ratio  of  the  quantity  of  heat  required  to  raise  a  given 
mass  of  the  substance  at  any  given  temperature  through  one 
degree,  to  the  quantity  required  to  raise  an  equal  mass  of 


Heat  as  a  Quantity  81 

water  through  the  same  range  of  temperature,  is  called  the 
specific  heat  of  the  substance  at  that  temperature. 

Specific  heat  is  a  ratio,  and  therefore  does  not  depend 
upon  any  particular  kind  of  units.  Numerically,  it  is  equal 
to  the  thermal  units  required  to  raise  a  unit  of  mass  of  the 
substance  through  i°  of  temperature,  and  is  sometimes  defined 
as  such. 

In  the  above  experiment  the  mass  in  each  is  i  Ib. 

2C 

.*.  specific  heat  of  turpentine  =  -^-  -  0-42 

If  2  Ibs.  of  iron  at  100°  C.  be  mixed  with  5  Ibs.  of  water 
at  10°  C.,  the  resulting  temperature  will  be  about  14°  C. 

To  raise  the  temperature  of  5  Ibs.  of  water  from   10°  to  14° 

requires  5X4  thermal  units  ; 
2  Ibs.  of  iron  cooling  86°,  i.e.  (100  -  14)°,  give  up  20  thermal 

units  ; 

20 
.*.  2  Ibs.  of  iron  cooling  ic  give  up  -^  thermal  units;   i.e.  0-24 

thermal  unit  nearly 

heat  required  to  raise  2  Ibs.  of  iron  i° 
.'.  specific  heat  of  .ron  =  heat  required  to  raise  2  Ibs.  of  water  .^ 

2O 


The  following  experiments  roughly  show  that  the  specific 
heats  of  substances  vary  :  — 

Take  discs  of  silver  (a  florin),  copper  (a  penny),  iron,  lead, 
same  size  and  thickness  ;  hold  them  at  equal  distances  from 
a  bright  fire  ;  in  one  minute  test  the  temperature  of  each  with 
a  thermometer.  Their  order  as  regards  temperature  is  —  lead, 
silver,  copper,  iron.  All  have  approximately  received  equal 
amounts  of  heat,  and  we  infer  that  the  specific  heat  of  iron, 
for  example,  is  greater  than  that  of  lead.  (To  make  the  experi- 
ment more  exact,  equal  masses  should  be  taken,  that  is,  the 
thicknesses  of  the  discs  should  vary  inversely  as  the  densities.) 

Weigh  into  three  separate  test-tubes  two  ounces  of  iron 
shot,  two  ounces  of  finely  granulated  lead,  two  ounces  of 

G 


82  Heat 

mercury  ;  place  the  test-tubes  in  a  beaker  of  boiling  water, 
and  let  them  remain  some  time.  Weigh  into  three  separate 
beakers  two  ounces  of  water,  and  observe  the  temperature. 
In  an  experiment  it  was  16°  C.  Remove  the  test-tubes  and 
pour  the  contents  (at  100°  C.)  into  the  separate  beakers  ;  stir, 
and  note  as  carefully  as  you  can  the  final  temperature. 

(a)  Mercury  and  water,  final  temperature  =  19° 

(b)  Zinc          „         „         „  „  =  23° 

(c)  Iron          „         „         „  „  =  25° 

/.  2  ozs.  of  mercury  cooling  81°  heat  2  ozs.  of  water  3° 
2      „      zinc  „        77°    „     2     „        „         7° 

2      „      iron  „        75°    „     2     „         „         9° 

T  o 

.'.  2  ozs.  of  mercury  cooling  i°  heat  2  ozs.  of  water  — 
2      „       zinc  „         i°      „     2      „         „      -L 

j   O 

2      ,,       iron  „         i°      „     2      „         „      — 

"<% 
.*.  specific  heats  of  water,  iron,  zinc,  mercury,  are  as  — 

i  -  A.  _L.  JL 
'  25  *  n  *  27 

If  m,  s,  /,  represent  the  mass,  specific  heat,  and  temperature 
of  a  substance  ;  and  M,  S,  T,  represent  the  mass,  specific  heat, 
and  temperature  of  water  (S,  of  course,  =  i);  and  if  0  be  the 
final  temperature  after  mixing  the  substance  and  the  water— 

ms(t-0)  =  MS(0-T) 
= 


t  -  0 


It  is  assumed,  in  the  above  calculations,  that  all  the  heat 
given  up  by  the  substance  is  used  in  heating  the  water. 

The  specific  heat  of  substances  can  be  roughly  determined 
by  using  a  beaker  or  a  thin  copper  vessel  as  the  calorimeter* 


•Heat  as  a  Quantity  83 

Solids  should  be  in  strips  and  wound  into  a  spiral  shape,  so  that 
they  may  the  more  readily  give  up  their  heat.  By  attaching 
them  to  a  fine  piece  of  thread  and  suspending  them  in  steam 
they  can  be  raised  to  100°  C.  The  temperature  of  a  weighed 
quantity  of  water  in  the  calorimeter  is  taken,  the  solid  is  rapidly 
introduced,  the  whole  stirred,  and  the  final  temperature  read. 

It  is  obvious  that  there  are  several  sources  of  error.  Heat  is 
lost  on  moving  the  solid,  part  is  used  in  heating  the  calori- 
meter and  thermometer,  and  part  is  lost  by  radiation. 

56.  Thermal  Capacity. — If  s  be  the  specific  heat  of  a 
substance  whose  mass  is  //*,  and  0  be  the  range  of  temperature 
through  which  it  is  heated — 

Then  to  raise  m  units  water  through  0°  would  req.  mB  therm,  units 
„      m  „  substance     ,,     6°          „        msO         „ 
„     ;//  „          „  ,,      i  „       ms  „ 

i.e.,  if  Q  be  the  quantity  of  heat — 

Q  =  msO 
~  is  called  the  thermal  capacity  of  the  substance  :  it  is  the  heat 

required  to  raise  the  temperature  of  the  substance  through 
one  degree. 

For  example,  the  quantity  of  heat  necessary  to  raise  8  Ibs. 
of  turpentine  through  15°  is — 

Q  =  8  X  0-42  x  15  -  50-4  thermal  units 
It  is  sometimes  inconvenient  or  unnecessary  to  separate  ;// 
and  s.  Thus  we  might  find  by  experiment  the  quantity  of  heat 
necessary  to  raise  a  mass  of  turpentine  through  15°.  Suppose 
the  quantity  was  50*4  thermal  units.  Then  to  raise  this  mass 
of  turpentine  through  i°  would  require  3*36  units. 

Then  3*36  units  would  be  the  mean  thermal  capacity  of  the 
substance  through  the  given  range.  If,  in  addition,  we  knew  that 

the  mass  was  8  Ibs.,  then  -^-  =  0^42  is  the  thermal  capacity 

of  unit  mass.     This  is  numerically  equal  to  the  specific  heat. 

The  expression,  "  thermal  capacity  of  unit  volume,"  is  also 
In  use  ;  it  is  the  product  of  the  specific  heat  and  the  density. 


84  Heat 

57.  Water  Equivalent.  —  The  thermal  capacity  of  a  sub- 
stance is  the  product  of  its  mass  and  specific  heat,  m  x  s.    It 
represents  the  number  of  thermal  units  required   to  raise  its 
mass  through  one  degree.     Now,  if  we  take  a  mass  of  water,  to, 
numerically  equal  to  ;//  X  ^,  then  it  will  take  the  same  quantity 
of  heat  to  raise  m  units  of  the  given  substance   through  one 
degree  as  it  takes  to  raise  co  units  of  water  through  one  degree. 

to  is  called  the  water  equivalent  of  the  substance,  seeing  that 
calculations  will  not  be  affected  if  we  imagine  the  substance 
removed,  and  CD  units  of  mass  of  water  substituted  for  it. 

58.  The  Method  of  Mixtures.—  Specific  heats  are  fre- 
quently determined  by  the  method  of  mixtures.     Easy  examples 
have  already  been  given. 

If  mlt  slt  and  /i  represent  the  mass,  specific  heat,  and  tem- 
perature of  a  given  body  ;  and  m.2,  s&  and  /2  the  mass,  specific 
heat,  and  temperature  of  a  second  body;  then,  if  they  be 
mixed  and  no  heat  enters  or  escapes,  and  if  the  heat  given  up 
by  one  be  wholly  used  in  heating  the  other,  they  finally  assume 
a  common  temperature,  0.  The  hotter  body  has  fallen  from 
/2  to  0,  and  the  colder  has  risen  from  /x  to  0.  The  former 
has  lost  /«2M/2  —  0)  units  of  heat  ;  the  latter  has  gained 
0*i-fi(0  —  *)  units.  These  are  equal. 

-  0) 


mlsl  4- 

also  ,   - 
>  '*  ~ 


-  A) 


If  there  be  three  substances,  we  have  — 

-  4)  -  w3j3(/3  -  6) 


^  +  m^  -f  mss3 
Jfi  ~  A) 


Heat  as  a  Quantity 


85 


The  heating  of  the  surrounding  vessel  and  the  heating  of 
the  thermometer  cannot  be  neglected  in  practice.  The  ex- 
periment is  carried  out  as  follows  : — 

59.  The  Calorimeter. — The  vessel,  a  cylinder  of  copper, 
E  (Fig.  41),  is  suspended  inside  a  similar  though  larger 
cylinder,  F,  by  threads,  in  order  to  prevent,  as  far  as  possible, 
loss  of  heat  by  conduction.  The  outside  of  the  former  and 
the  inside  of  the  latter  are  highly  polished,  the  aim  being 


FIG.  41. 

to  ensure  that,  if  any  heat  be  radiated  by  the  inside  vessel, 
as  much  as  possible  shall  be  reflected  by  the  outside  vessel 
The  vessels  are  placed  inside  a  wooden  box,  G,  and  are  packed 
in  felt  to  prevent  variation  in  the  temperature  of  the  sur- 
rounding space  from  affecting  the  calorimeter.  A  weighed 
quantity  of  water,  m,  is  placed  in  E,  and  the  temperature  / 
is  read  from  the  thermometer  K,  which  is  secured  to  the  stand 
H.  Neglecting  the  water  equivalent,  we  have  the  simple 
arrangement  of  §  55. 


86  Heat 

To  heat  the  substance,  say  a  small  cylinder  of  iron  or 
copper  (if  possible  the  solid  should  be  in  spiral  form,  the 
better  to  give  up  its  heat),  it  should  be  exposed  for  some  time 
to  steam  under  ordinary  atmospheric  pressure.  This  can  be 
done  in  the  usual  steam-jacket  modified  as  follows  :  A  closed 
copper  vessel,  A,  has  a  cylinder,  B,  passing  through  it ;  steam 
enters  from  a  boiler  by  the  lower  side  tube,  and  escapes  by 
the  upper  side  tube  into  a  condenser.  The  central  cylinder 
is  closed  at  the  top  by  a  cork.  A  thermometer,  P,  passes 
through  the  cork,  and  the  solid,  M,  is  suspended  by  a  very 
fine  thread  that  also  passes  through  the  cork.  The  heater 
revolves  around  D,  and  ordinarily  the  bottom  of  B  is  closed 
by  moving  A  so  that  it  rests  upon  the  solid  board  of  the 
stand.  The  enclosure  is  then  at  a  temperature,  /2>  very  near 
100°  C.  When  A  is  in  the  position  of  the  figure,  it  is  over 
a  circular  aperture  cut  in  the  board  whose  diameter  is  equal 
to  that  of  B. 

To  transfer  the  solid  to  the  calorimeter  with  as  little  loss 
of  heat  as  possible,  the  following  combined  arrangement  is 
made : — 

The  calorimeter  box  slides  on  along  two  wooden  rails,  and, 
by  lifting  the  door  L,  can  be  .quickly  slid  into  position. 
When  the  temperature  of  the  heater  is  constant,  L  is  raised,  the 
calorimeter  is  slid  beneath  the  hole  in  the  stand,  the  heater  is 
brought  over  the  hole,  the  solid  quickly  lowered,  and  the 
calorimeter  is  then  replaced  in  its  first  position,  stirred  by  its 
thermometer,  and  the  highest  point  reached  by  the  ther- 
mometer noted. 

In  order  to  correct  for  the  heat  that  raises  the  temperature 
of  the  calorimeter  and  the  thermometer,  the  water  equivalent 
is  first  determined.  This  is  done  by  (a)  weighing  the  calori- 
meter, placing  water  in  the  calorimeter;  and  (b)  weighing 
again,  noting  the  temperature,  and  then  adding  a  quantity  of 
water  at  about  30°,  stirring,  and  noting  the  final  temperature ; 
and  (c)  weighing  again,  b  —  a  gives  the  mass  m  of  water  at 
/°,  the  temperature  of  the  room ;  c  —  b  gives  the  mass  of  water 
mi  added  at  t°.  Let  0°  be  the  final  temperature.  Since  the 
heat  from  a  mass  ml}  in  cooling  from  t°  to  0°,  is  used  in  heating 


Heat  as  a  Quantity  87 

a  mass  m  from  ;°  to  6°,  and  also  in  heating  the  calorimeter 
and  thermometer  from  /°  to  0°,  we  have  — 


where  <o  is  the  water  equivalent. 


The  water  equivalent  could  be  calculated  if  we  knew  the 
mass  of  the  calorimeter,  //,,  the  specific  heat  of  its  substance,  o-; 
and  the  mass,  ftj,  and  specific  heat  of  the  thermometer,  o-lt 
Then— 


The  determination  by  actual  experiment,  conducted  as  far 
as  possible  under  similar  conditions  as  to  temperature  as  in 
determining  the  specific  heats,  is,  however,  to  be  preferred. 

In  determining  the  specific  heat  of  a  substance  of  mass 
mzi  if  s.2  be  the  specific  heal,  and  /2  the  temperature  to  which 
it  is  heated,  then,  if  m  be  the  mass  of  water  in  the  calorimeter 
at  /°,  and  <o  be  the  water  equivalent,  and  0  the  final  tempera- 
ture, we  obtain,  as  in  §  58— 

;;/2j2(/s  -  0)  =  (m  -f  <o)(0  -  /) 


m,(t,  -  6) 

In  determining  the  water  equivalent,  the  final  temperature 
should,  as  far  as  possible,  approximate  to  the  final  temperature 
in  determining  the  specific  heats.  There  will  be,  despite  these 
precautions,  a  loss  due  to  radiation  in  determining  the 
specific  heat  of  a  substance.  This  is  reduced,  if  Rumford's 
method  be  followed,  of  arranging  the  experiment  so  that  the 
original  temperature  of  the  water  in  the  calorimeter  be  as  far 
below  the  temperature  of  the  room  as  the  final  is  above  it. 
During  the  first  part  of  the  experiment  the  calorimeter  will 
receive  heat  from  the  room  ;  during  the  latter  part  it  will  give 
up  heat  to  the  room  ;  these  two  quantities  are  nearly  equal. 

The  specific  heat  of  liquids  can  be  found  by  enclosing  them 
in  a  thin  glass  vessel  of  known  mass  and  known  specific  heat 


88 


Heat 


— preferably  the  vessel  should  be  hermetically  sealed— heating 
as  before,  and  allowing  for  the  heating  of  the  glass;  or  by 
using  the  liquid  instead  of  water  in  the  calorimeter,  and  heating 
and  dropping  in  a  mass  of  metal  of  known  specific  heat. 

For  example,  if  m  be  the  mass  of  the  solid,  s  its  specific 
heat,  m'  and  s'  mass  and  specific  heat  of  the  liquid — 

ms(t  -0)  =  m's'(e  -  /')  +  o,(0  -  t) 

where  all  is  known  save  /. 

It  is  supposed  in  all  cases  that  no  chemical  action  takes 

place,  and  that  the  substance  does  not  dissolve  ;  for  example, 

for  substances  soluble  in  water  some  other  liquid  in  which 

they  are  insoluble  must  be  used. 

The  apparatus  Fig.  41  is  practically  that  of  Regnault 
60.  Latent  Heat  of  Fusion. — By  mixing  i  Ib.  of  ice  at 

o°  with  i  Ib.  of  water  at  80°  C,  we  obtain  2  Ibs.  of  water  at  o° ; 


FIG.  42.  FIG.  43. 

that  is,  80  thermal  units  are  needed  to  change  i  Ib.  of  ice  at 
o°  into  water  at  o°  This  is  called  the  latent  heat  of  fusion. 
It  will  be  discussed  later,  and  is  introduced  here  in  order  that 
the  following  calorimeters  may  be  understood. 

61.  Lavoisier  and  Laplace's  Calorimeter. — The  calori- 
meter A  is  placed  inside  a  vessel,  B,  from  which  it  is  separated 
by  melting  ice  (Figs.  42  and  43).  B,  in  its  turn,  is  inside  an 


Heat  as  a  Quantity  89 

outer  vessel,  and  is  also  packed  in  ice.  The  result  is  that  the 
ice  between  A  and  B  cannot  be  melted  by  any  heat  from  the 
outside.  The  mass  M,  whose  specific  heat  (s)  is  to  be  deter- 
mined, is  raised  to  a  known  temperature,  /,  and  is  placed  in 
A,  and  the  lids  are  replaced.  M,  cooling  to  o°  C,  melts  ice 
between  A  and  B.  The  water  drains  off  by  D,  and  is  weighed. 
If  m  be  the  mass  of  ice  at  o°  melted  to  water  at  o°,  then — 

MJ/  =  Som 

Som 

'~  MT 

The  defects  are  grave :  it  is  practically  impossible  to  work 
with  small  amounts,  and  even  with  large  masses  it  is  impossible 
to  drain  off  the  whole  of  the  water. 

62.  Bunsen's  Calorimeter.— Ice  floats  on  water.  It  is 
therefore  less  dense,  and,  in  melting,  the  water  formed  occupies 
a  less  volume  than  the  ice. 

A  glass  vessel,  ABC,  has  a  thin  glass  test-tube,  D,  fused 
into  it  (Fig.  44).  A  is  filled  with  distilled  water  that  has  been 


FIG.  44. 

thoroughly  boiled,  by  a  method  that  is  practically  that  of  filling 
a  thermometer.  Then  mercury  is  allowed  to  enter  by  the  opening 
at  C,  and,  by  tilting,  it  is  arranged  that  the  mercury  rises  to 
b  f,  the  water  being  above.  The  mercury  must  be  freed  from 
water  and  air.  The  narrow  tube  d  e  is  inserted  into  C  by  a 
cork,  F,  the  mercury  rises  and  flows  along  d  e.  de  has 
been  carefully  calibrated  along  its  horizontal  part.  The  whole 
apparatus,  which  must  be  free  from  all  bubbles  of  air,  is  now 


90  Heat 

placed  in  melting  ice  or  snow,  and  left  until  its  temperature 
falls  to  zero.  It  is  now  necessary  to  freeze  part  of  the  water 
surrounding  the  tube  D.  The  water,  being  free  from  air,  is 
somewhat  difficult  to  freeze.  The  freezing  can  be  readily 
accomplished  by  passing  through  D  a  current  of  alcohol 
previously  cooled  by  a  mixture  of  ice  and  salt. 

The  whole,  save  the  tube  de,  is  surrounded  with  melting  ice. 

By  moving  the  cork  F  it  is  arranged  that  the  mercury 
reaches  a  fixed  point,  e;  a  known  quantity  of  water  at  a  known 
temperature  is  poured  into  the  test-tube  D,  and  a  cork  is 
inserted.  The  water,  on  cooling  to  o°,  melts  some  of  the  ice 
surrounding  D,  there  is  a  consequent  contraction,  and  the 
column  moves  to  s,  say.  .$•  and  e  can  be  carefully  observed 
with  a  telescope.  Let  se  be  //  divisions;  then  a  mass  of  water 
m  has  cooled  from  f  to  o°. 

.*.  every  unit  of  heat  is  indicated  by  ^-  divisions 

mt 

mt  , 
.*.  i  division  represents  —  thermal  units 

In  order  to  use  the  instrument,  water  to  about  the  level  offg 
is  introduced,  and  left  until  it  gets  to  freezing  point.  The  cork  is 
adjusted  so  that  the  mercury  is  at  e.  The  body  whose  specific 
heat  is  required  is  weighed,  heated  to  a  given  temperature, 
dropped  into  the  test-tube  D,  and  the  cork  inserted.  The  heat 
is  given  up  to  the  lower  strata  of  water,  whose  temperature  rises, 
but  as  the  maximum  density  is  at  4°,  if  the  mass  be  calculated 
so  that  the  temperature  does  not  rise  above  4°,  then  the  heated 
water  keeps  at  the  bottom.  In  cooling,  it  melts  the  ice,  and 
sends  the  mercury  along  n'  divisions,  say.  Then,  if  m'  be  the 
mass,  s'  its  specific  heat,  and  t'  the  temperature,  the  n'  divisions 

indicate  -   -  thermal  units  obtained  by  cooling  the  mass  m1 


n' .  mt 

•••>"'=  — 

,      n1     mt 
=  7i'~rit' 
The  apparatus,  if  once  set  up  and  kept  in  ice  after  each 


Heat  as  a  Quantity  91 

experiment,  can  be  used  several  times.  It  is  only  necessary 
at  each  experiment  to  see  that  the  index  is  at  e. 

63.  The  Method  of  Cooling.  —  Accurate  experiments 
by  this  method  are  difficult,  but  the  principle  is  important  It 
depends  upon  the  fact  that  the  rate  at  which  a  body  loses 
heat  depends  upon  (i)  its  surface,  and  (2)  its  temperature 
above  surrounding  substances  (§§  169,  170). 

Suppose,  for  example,  that  a  globe  containing  water  at  50° 
be  suspended  in  the  middle  of  a  room  whose  walls  are  at  o°. 
Then,  neglecting  any  slight  difference,  the  rate  at  which  the 
globe  loses  heat  depends  upon  the  difference  of  temperature 
(50°  —  o°)  and  upon  the  surface  of  the  vessel.  Of  course, 
soon  the  difference  will  be  49°,  then  48°,  etc.,  and  thus  the 
rate  will  change. 

Suppose  the  water  be  removed,  and  turpentine  inserted 
at  50°.  The  loss  of  heat  would  be  the  same,  as,  it  depends 
upon  the  differences  of  temperature  and  the  surface  of  the 
vessel,  and  not  upon  the  substance  the  vessel  contains. 

Note  the  time  it  takes  the  vessel,  filled  with  water,  to  cool 
from  40°  to  39°.  Let  this  be  n  seconds.  Then  note  the 
time  it  takes  to  cool  from  40°  to  39°  when  the  vessel  is  filled 
with  turpentine.  Let  this  be  n^  seconds. 

The  quantities  of  heat  (x  and  x^  that  have  escaped  in  the 

x       n 
two  cases  are  proportional  to  the  times;  that  is,  - 

xl     n± 

The  calorimeter  and  thermometer  lose  heat  in  both  cases, 
and  therefore  their  water  equivalent,  o>,  must  be  known. 

A  mass  of  water,  m,  whose  specific  heat  is  i,  has  cooled  i° 
and  lost  m  units  of  heat  ;  /.  total  loss  =  m  +  <o 

A  mass  of  turpentine,  ml}  whose  specific  heat  is  s^  has  cooled 
i°  and  lost  m^  units  of  heat  ;  .'.  total  loss  =  m^  +  o> 

x         m  -f-  o>         n 


;;/ 


92  Heat 

For  any  other  liquid,  whose  specific  gravity  is  sa,  we  shall 
have  a  similar  expression.  Therefore  specific  heats  are  readily 
compared. 

The  details  of  the  experiment  can  be  carried  out  as 
follows : — 

(a)  The  calorimeter  is  a  thin  brass  vessel  with  a  lid;  a 
sensitive  thermometer  passes  through  a  cork  in  the  lid. 

(b)  The  water  equivalent  must  be  determined  as  in  §  59. 

(c)  The   experiments   answer  best  with  liquids  (if  solids 
be  used,  they  must  be  pulverized).      They  should  nearly  fill 
the  calorimeter,  and  equal  volumes  should  be  used  in  each 
case. 

(d)  The  liquids  should  be  heated  to  a  convenient  tempera- 
ture, poured  into  the  calorimeter,  and  then  the   calorimeter 
placed  in  a  bath  at  that  temperature,  to  ensure  that  both  vessel 
and  liquid  are  at  the  same  temperature.     Knowing  the  mass 
of  calorimeter  and  thermometer,  a  second  weighing  with  the 
liquid  at  the  end  of  the  experiment  will  give  the  mass  of  the 
liquid. 

(e)  The  results  are  conveniently  recorded  on  squared  paper, 
the  ordinates  representing   times,  the  abscissae   temperatures. 
Join  the  points  by  a  free  curve.     The  first  part  of  the  curve 
should  be  disregarded ;  for  example,  if  the  initial  temperature 
be    100°,   a   uniform    condition  of  temperature  will   only   be 
attained  about  95°. 

In  comparing  specific  heats,  select  the  times  of  cooling  for 
the  same  range  of  temperature  in  each,  say  from  60°  to  59°,  or 
40°  to  39°.  If  one  of  the  liquids  be  water,  then  the  specific 
heat  of  the  other  liquid  can  be  determined  for  various  falls 
of  5°  at  first,  then  4°,  3°,  etc.,  as  the  temperature  sinks ;  the 
mean  specific  heat  can  be  deduced  for  each  range,  and  for 
a  final  result  the  mean  of  all  these  values  can  be  taken. 

64.  True  Specific  Heat. — The  specific  heats  found  have 
in  all  cases  been  the  average  specific  heats  between  certain 
temperatures.  There  is,  however,  no  reason  why  we  should 
assume  that  this  is  the  specific  heat  at  every  temperature ;  that, 
for  example,  the  specific  heat  of  iron  at  o°  =  the  specific  heat 
of  iron  at  100°,  or  that  the  specific  heat  of  water  at  o°  is  the 


Heat  as  a  Quantity 


93 


same  as  the  specific  heat  of  water  at  any  other  temperature 
— that  must  be  a  subject  for  experiment. 

Suppose  we  mark  on  OX  (Fig.  45),  at  equal  distances, 
numbers,  to  indicate  degrees  of  temperature,  and  at  each 
degree  draw  a  perpendicular,  representing  the  number  of 
units  of  heat  required  to  raise  the  temperature  of  a  given  sub- 
stance from  o°  to  that  degree.  Thus  3?  measures  the  number 
of  units  of  heat  required  to  raise  the  temperature  from  o°  to 


2«  3' 

TEMPERATURE 

FIG.  45- 

3°,  4Q  from  o°  to  4° ;  then  ^Q  =  amount  required  to  raise 
the  temperature  from  3°  to  4°,  rR  from  3°  to  5°.  Joining 
the  points,  O,  P,  Q,  R,  etc.,  we  get  a  curve  or  line.  If  twice 
^Q  =  rR  =  etc.,  and  if  3?,  4Q,  5R  be  some  multiple  of  the 
heat  required  to  raise  the  temperature  from  o°  to  i°,  we  shall 
have  a  straight  line,  O  P  Q  R. 

r^,  .-  quantity  of  heat 

The    specific   heat    numerically  =  — -    —rj- 

range  of  temperature ' 

i.e.  the  mean  specific  heat  from  3°  to  4° 

=  4Q-3P  =  ?Q 
4-3  i 

And  since  2(^Q)  =  HR,  the  specific  heat  is  constant  at  all 
temperatures. 

When  experiments  are  carefully  conducted,  it  is  found  that 
the  true  position  of  the  points  is  above  P,  Q,  R,  so  that  a 
curve,  O  P'  Q'  R',  is  formed. 


94  Heat 

Thus  to  raise  i  Ib.  of  water  from  o°  to  20°  requires  20*01 
units ;  from  o°  to  40°,  40-05  units ;  from  o°  to  60°,  60-14 
from  o°  to  1 00°,  100*5  units.     From  this  we  obtain — 


Between  o°  and  20° 


„    40 


40°  „  60 
60°  „  80 
80°  „  100 


Average  specific  heat 
of  water. 

•OOO5 

'OO20 

o 


•0045 
•OO70 
'OIIO 


The  specific  heat  of  water  rises  with  the  temperature  (the 
same  is  true  of  other  substances).  The  increase  is  very  small, 
and  can  be  disregarded  in  all  save  the  most  exact  experiments 
and  calculations.  It  is  introduced  to  show  that  the  specific 
heat  of  a  substance,  as  in  the  case  of  the  coefficient  of  dilata- 
tion, is  not  a  constant,  and  cannot  be  exactly  calculated  from 
knowing  the  specific  heat  from  o°  to  1°.  The  above  figures, 
for  water,  are  due  to  Regnault,  who  states  that  the  specific 
heat  of  water  at  any  temperature  can  be  calculated  from  the 
formula — 

Specific  heat  =  i  4-  o'oooo4/  +  o'oooooo9/2 

At  any  temperature  is  used  in  the  sense  of  §  33.  Analogous 
to  the  construction  in  Fig.  31,  the  specific  heat  at  any 
temperature  will  be  the  tangent  at  the  point  that  indicates 
the  given  temperature. 

The  general  result  is  that  the  specific  heat  of  substances 
rises  with  the  temperature. 

65.  High  Specific  Heat  of  Water.— Of  all  known  sub- 
stances water  has  the  greatest  specific  heat.  One  thermal 
unit  would  raise  i  Ib.  of  water  through  i° ;  but  the  tempera- 
ture of  i  Ib.  of  iron,  with  the  same  quantity  of  heat,  would 
be  raised  9°,  or  9  Ibs.  would  be  raised  i°.  Similarly,  the  same 
quantity  of  heat  would  raise  the  temperature  of  30  Ibs.  of 
mercury  i°,  or  4-2  Ibs.  of  air  i°.  In  the  case  of  gases  it 
becomes  more  striking  if  we  compare  by  volumes.  The  heat 
required  to  raise  i  cubic  foot  of  water  i°  would  raise  4-2  times 
the  mass  of  air  i°  ;  but  as  water  is  770  times  heavier  than  air* 


Heat  as  a  Quantity  95 

it  would  therefore  raise  770  x  4*2  =  3234  cubic  feet  of 
air  i°. 

"  The  vast  influence  which  the  ocean  must  exert  as  a  mode- 
rator of  climate  here  suggests  itself.  The  heat  of  summer  is 
stored  up  in  the  ocean,  and  slowly  given  out  during  winter. 
This  is  one  cause  of  the  absence  of  extremes  in  an  island 
climate." 

Other  examples  will  be  given  in  the  chapter  on  climate. 

The  high  specific  heat  of  water  is  utilized  in  heating  build- 
ings by  hot  water,  and  in  railway  foot-warmers. 

66.  Specific  Heat  of  Gases.—  The  specific  heat  of 
gases  can  be  calculated  — 

(1)  When    the   pressure    is    kept    constant,   the   volume 
varying  :  this  is  the  specific  heat  at  constant  pressure.     Or  — 

(2)  When  the  volume  is  kept  constant  and  the  pressure 
varies  :  this  is  the  specific  heat  at  constant  volume. 

In  determining  (i),  the  principle  is  to  allow  a  mass,  ;//,  of 
a  gas  to  pass  through  a  spiral  in  a  bath  of  boiling  oil.  It  there 
attains  the  temperature  f.  It  then  enters  a  specially  con- 
structed calorimeter  containing  a  mass,  #/',  of  water  at  f°9  so 
arranged  that  the  gas  has  to  pass  through  several  spirals 
surrounded  by  the  water  of  the  calorimeter,  thus  giving  up  its 
heat  to  the  water  in  the  calorimeter,  before  escaping  at  the 
temperature  (/"°)  indicated  by  the  water  of  mass  m'  and  at 
atmospheric  pressure  ;  this  is  the  pressure  the  gas  has  been 
subject  to  throughout  the  experiment.  Therefore— 

ms(t-t")  =  m'(t"  -  t) 


s= 

Elaborate  precautions  must  be  taken  in  order  that  the 
above  conditions  may  be  fulfilled  ;  particulars  of  the  experi- 
ment will  be  found  in  Jamins'  "  Cours  de  Physique." 

The  specific  heat  of  gases  at  constant  volume  is  difficult 
to  estimate  directly.  It  is  calculated  from  the  known  relation 
that- 

specific  heat  of  gases  at  constant  pressure       C 
specific  heat  of  gases  at  constant  volume        c 


96  Heat 

For  air,  hydrogen,  and  many  other  gases  a  mean  value  of 
K  is  i '41 ;  for  carbon  dioxide  it  is  1*29.  This  will  be  referred 
to  again  (consult  Index). 

The  specific  heats  are  obtained  as  before  by  comparing 
with  equal  mass  of  water. 

SPECIFIC  HEATS  OF  GASES  AT  CONSTANT  PRESSURE. 
WATER  =  i. 

Equal  weights.  Equal  volumes. 

Air  ...  0-2374  0-2374 

Oxygen  ...  0-2175  •••  °'2405 

Hydrogen  ...  3-4090  0-2359 

Nitrogen  ...  0*2438  ...  0-2370 

Chlorine  ...  0-1219  ...  0-2962 

Carbonic  acid  ...  0-2169  ...  0-3307 

Hydrochloric  acid  0*1845  •••  °'2333 

Ammonia  ...  0-5083  ...  0-2966 

The  second  column  gives  the  units  of  heat  required  to 
raise  equal  volumes  of  the  substances  one  degree,  compared 
with  the  amount  required  to  raise  an  equal  volume  of  air  one 
degree,  under  similar  conditions  of  temperature  and  pressure, 
the  air  being  compared,  as  in  column  i,  to  an  equal  mass  of 
water. 

67.  Effect  of  Temperature  and  Pressure.— The 
specific  heats  of  gases,  according  to  Regnault,  do  not  vary 
with  the  pressure  ;  i.e.  the  same  results  were  obtained,  whether 
the  experiments  were  conducted  under  a  pressure  of  i  or  2 
atmospheres,  provided  that  the  pressure  was  constant  through- 
out each  experiment,  and  in  the  case  of  air  and  gases  not 
easily  condensed,  the  specific  heat  is  practically  constant  at 
all  temperatures. 

In  the  case  of  pressure,  if  s  be  the  mean  specific  heat  of 
a  substance,  8  its  density,  then  sB  =  thermal  capacity  of  unit 
volume  =  K. 

.*.  K  (at  a  pressure  p)  :  K  (at  a  pressure  /')  =  sS  :  s$  =  8  :  8' 

Or,  the  thermal  capacity  of  unit  volume  varies  as  the  density 
of  the  gas. 


Heat  as  a  Quantity  97 

The  fact  that  the  specific  heat  of  air  is  unaffected  by 
temperature  and  pressure  points  out  the  value  of  air  as  a 
thermometric  substance,  as  equal  quantities  of  heat  will  pro- 
duce equal  expansion  on  all  parts  of  the  scale,  and  therefore 
an  air- thermometer  will  nearly  agree  in  all  parts  of  the  scale 
with  an  absolute  scale  of  temperature. 

The  variations  in  the  specific  heats  of  gases  must  be  very 
small,  or  they  would  not  have  escaped .  a  careful  experimenter 
like  Regnault.  Joly,1  however,  finds  that  the  specific  heats  of 
gases  at  constant  volume  increase  with  the  pressure  between 
the  limits  of  his  experiments.  He  has  obtained  the  following 
numbers  : — 

Gas,  and  Pressure  in  Specific  heat 

mean  temperature.  atmosphere?.  Density.  (constant  volume). 

Air  (50°  C.)  13-56  0-01428  0-17193 

23-35  ...  0-02459  ...  0-17223 

Carbon  dioxide  (55°  C.)  1 2 -io  ...  0*01979  ...  0-16922 
21-66  ...  0-03780  ...  0-17386 

68.  Specific  Heat  changes  with  Change  of  State. — 

The  specific  heat  of  water  in  the  liquid  state  is  unity,  but  that 
of  ice  is  only  about  one-half,  while  the  specific  heat  of  water 
vapour  is  also  about  one-half. 

These  changes  are  illustrated,  for  some  substances,  in  the 
following  table  :  — 

MEAN  SPECIFIC  HEATS. 

Gas 
Solid.  Liquid.  (constant  pressure). 

Water 0-504       ...       1*000       ...       0-477 

Mercury  ...     0-031        ...       0*033  0*015 

Lead      ...         ...     0*031       ...       0*040 

Even  when  the  substance  remains  as  a  solid,  the  specific 
heat  varies  with  any  change  in  its  physical  condition.  For 
example,  calcium  carbonate  has  a  specific  heat  of  0*2085  as 
aragonite,  0*2148  as  chalk,  and  0*2158  as  marble. 

1  J«  J°1>T>  M.A.,  "  Philosophical  Transactions,"  1891. 

H 


98 


Heat 


Similar  changes  are  shown  in  the  various  allotropic  forms  of 
sulphur. 

This  is  important  in  discussions  relative  to  the  mechanical 
equivalent  of  heat,  and  also  in  respect  to  the  nature  of  heat. 
It  is  a  subject  for  experiment,  for  example,  whether  the  specific 
heat  of  a  bar  of  iron  is  exactly  equal  to  the  same  or  an  exactly 
similar  bar  twisted  into  a  spiral ;  or  whether  a  piece  of  lead, 
a  few  hours  after  solidifying,  will  have  the  same  specific  heat 
now  as  at  the  end  of  a  month,  when  its  molecules  will  have 
had  time  to  occupy  more  permanent  positions,  or  after  subject- 
ing it  to  great  pressure  in  an  hydraulic  press.  In  the  case 
of  lead  or  steel  no  difference  has  been  detected  in  experi- 
ments, but  no  assumptions  must  be  made  for  these  or  other 
substances. 

69.  Dulong  and  Petit's  Law. — The  product  of  the 
specific  heats  and  the  atomic  weights  of  solid  substances  is  a 
constant,  which  is  approximately  6-4;  or,  the  specific  heat  of 
an  element  varies  inversely  as  the  atomic  weight. 

This  is  seen  from  the  following  table  : — 


Specific  heat. 
(*) 

Atomic  weight. 
<*) 

Atomic  heat. 
(ax*) 

Aluminium 
Antimony 
Bismuth  ... 

0-2143 
0-0513 
0-0308 

27-4 
122 

210 

5-9 
6'3 
6'5 

Copper    .  .  . 

0-0939 

63-5 

6-6 

Iron 

0-1138 

56 

6-4 

Platinum 

0-0324 

197*5 

6-4 

Zinc 

0-0950 

65-2 

6-2 

Another  form  of  stating  the  same  law  is  that  the  same 
quantity  of  heat  is  required  to  raise  the  temperature  of  an  atom 
of  all  simple  bodies  through  the  same  range  of  temperature. 

Carbon,  silicon,  and  boron  for  some  time  seemed  to  be 
marked  exceptions,  but  the  specific  heats  of  these  substances 
rise  with  the  temperature.  At  high  temperatures  they  approxi- 
mate to  the  constant, 


Heat  as  a  Quantity 


99 


Temperature. 

Specific  heat. 

Atomic 
weight. 

Atomic 
heat. 

Carbon 

33° 

0*1318 

12 

1-6 

(diamond) 

247° 

0-3026 

3'6 

985° 

Q'459 

5'5 

Silicon 

22° 

0-1697 

28 

47 

(crystalline) 

232° 

0-203 

57 

And  it  is  probable  that  the  law  is  true  at  very  high  temperatures. 
If  the  specific  heats  of  simple  gases  (equal  weights)  be 
multiplied  by  the  relative  densities  (hydrogen  =  i),  then  a 
product  is  obtained  which  represents  the  thermal  capacity  of 
equal  volume.  This  is  found  to  be  fairly  constant. 


Oxygen       
Hydrogen  
Nitrogen     ... 

Specific  heat. 

Atomic  weight. 

Product. 

0-2175 
3-410 
0-2438 

16 

i 
14 

3-48 
3'4I 
3'4I 

Assuming  Avogadro's  law,  that  equal  volumes  of  gases 
under  the  same  conditions  as  to  temperature  and  pressure 
contain  the  same  number  of  molecules,  we  conclude  that  the 
thermal  capacity  of  all  molecules  of  the  simple  gases  is  a 
constant  The  molecular  weight  in  the  above  gases  is  twice 
the  atomic  weight. 

The  student  will  remember  that  the  specific  heats  of  the 
solids,  in  Dulong  and  Petit's  law,  are  deduced  from  bodies 
in  the  solid  state.  It  is  possible,  when  definite  and  direct 
experimental  knowledge  is  gained  of  solid  oxygen,  hydrogen, 
etc.,  that  the  constant  may  approach  6-4. 

When  chemical  compounds  of  the  same  class  are  examined, 
a  similar  law  is  found ;  thus — 


Oxide  of  lead  (PbO) 
Oxide  of  merrury  (HgO) 
Oxide  of  copper  (CuO)  ... 

Specific  heat. 

s» 

Molecular  weight. 
M. 

Product. 
SX  M. 

0-0509 
0-0518 
0-1420 

223 
216 
79'5 

II'4 
1  1  '2 

ii'3 

ioo  Heat 

For  the  sulphides  the  product  is  18-9,  and  for  the  ses- 
quioxides  the  product  is  27-1.  The  carbonates  (CaCO3, 
BaCO3)  give  a  mean  number  of  21-8. 

69a.  Siemens's  Water  Pyrometer. — This  is  a  good 
illustration  of  specific  heats.  The  calorimeter  is  a  copper 
vessel  made  with  a  double  casing  of  copper.  The  space  between 
the  casings  is  filled  with  felt;  this  is  placed  inside  another 
copper  vessel  to  prevent  radiation.  A  pint  of  water  (568 
grams)  is  placed  in  the  calorimeter,  and  a  thermometer, 
graduated  in  the  ordinary  way,  is  inserted.  A  brass  scale  is 
movable  along  the  thermometer  scale,  and  is  graduated  with 
50  divisions  to  one  degree  of  the  thermometer  scale.  The 
zero  of  the  sliding  scale  is  fixed  at  the  temperature  indicated 
by  the  thermometer. 

A  copper  cylinder  weighing  137  grams  (if  of  iron,  it  weighs 
112  grams,  and,  if  platinum,  402*6  grams)  is  placed  in  the 
furnace,  and  left  for  a  few  minutes  until  it  attains  the  tempera- 
ture of  the  furnace ;  it  is  then  quickly  dropped  into  the 
pyrometer,  and  the  rise  in  temperature  is  read  off  on  the  sliding 
scale,  which  gives  fifty  times  the  rise  in  temperature  in  degrees. 

This  rise,  as  shown  on  the  sliding  scale,  added  to  the 
temperature  of  the  water  at  the  commencement  of  the  experi- 
ment, gives  the  temperature  of  the  furnace. 

The  weight  of  the  cylinder  is  selected  so  that  its  thermal 
capacity  for  one  degree  is  equal  to  one-fiftieth  that  of  a  pint 
of  water  +  the  water  equivalent  of  the  calorimeter.  In  this 
calorimeter  the  water  equivalent  is  82  grams;  that  is,  water 
4-  calorimeter  are  equivalent  to  650  grams  of  water.  The 
thermal  capacity  of  the  copper  should  be  one-fiftieth  of  this ; 
that  is,  equivalent  to  13  grams  of  water.  Therefore  the  weight 
of  the  cylinder  should  be  13  -r-  0-095  =  X37  grams.  As  the 
cylinder  loses  weight  with  use,  correction,  of  course,  must  be 
applied ;  for  example,  when  its  weight  is  only  130  grams,  its 
thermal  capacity  is  12*35  thermal  units;  that  is,  a  rise  of  one 
degree  in  the  temperature  of  the  water  indicates  an  excess 
of  650  4-  12-35  =  52-6°  in  the  temperature  of  the  furnace. 

(This  particular  pyrometer  is  described  in  Baird  and 
Tatlock's  catalogue.) 


Heat  as  d  Quantity- 


101 


WORKED  EXAMPLES. 

i.  20  grams  of  iron  at  98°  C.  (specific  heat,  0*119)  are  immersed  in 
80  grams  of  water  at  10°  C.  contained  in  a  copper  vessel  whose  mass  is 
15  grams:  find  the  resulting  temperature,  the  specific  heat  of  copper 
being  0*095. 

Let  6  be  the  resulting  temperature. 

(1)  The  iron  gives  up  20  X  0*119(98  —  6)  thermal  units 

(2)  The  water  receives  8o(0  —  10)  ,,          (a) 
The  calorimeter    ,,        15  x  0*095(6  —10)        „          (8\ 

(a  +  6)  =  (0  -  10)  (80  +  1-425) 
Heat  given  up  in  (i)  =  heat  absorbed  in  (2) 

.-.  2*38(98-6)  =  81-425(6-  10) 
,*.  6(81-425  +  2-38)  =  98  X  2*38  +  814*25 
=  1047-49 


2.  Determine  the  specific  heat  of  zinc  (Day's  Examples). 

(1)  Calorimeter  of  brass  weighs  55  *H  grams 

(2)  Calorimeter  +  water    ,,  5I7'5O      ,, 

(3)  Zinc  in  pieces  „  293*65      ,, 

It  is  placed  in  a  brass  cage  weighing      8*48       ,, 

(4)  Thermometer:  the  glass  weighed  1*27      ,, 

,,  mercury    „  7*62      ,, 

Specific  heats  given  :  brass  =  0*094  ;  mercury  =  0*033  >  glass  =  0-198. 
Zinc  in  cage  was  immersed  at  a  temperature  of  99-1  1°  C.  ;  the  water  was 
initially  at  o°  C.,  its  final  temperature  5*22°  C. 

The  calorimeter  contains  462*39  grams. 

The  water  equivalent  of  calorimeter  and  thermometer  =  o> 

=  (55'H  x  °'°94)  +  (7'62  X  0-033)  +  (I>27  X  0*198) 

5*1832         +        0*2515       +        0*2515 
=  5-686 
Let  s  be  the  specific  heat  of  zinc. 

(1)  Then  zinc  and  cage  lose  (293-65^+8  -48x0-094)  (99*11-5-22)  thr.  un. 

(2)  The  calorimeter  gains  5-22(462*39+5*686)  thermal  units. 

5*22  X  468*076 
293-65*  =  L-^?89--      -°'/97 

_  26*023  -  0*797  _  25*226 
293-65    ~  293-65 
=  0*086 

EXAMPLES.    V. 


i.  A  ball  of  copper  at  98°  C.  is  put  into  a  copper  vessel  containing 
Ibs.  of  water  at  15°,  and  the  temperature  of  the  water,  ball,  and  vessel 


IO2 


after  the  experiment  is  21°  C.  ;  the  weight  of  the  ball  is  I  lb.,  and  the 
specific  heat  of  copper  O'O95  :  find  the  weight  of  the  copper  ball. 

2.  A  copper  ball  weighing  61bs.,  taken  out  of  a  furnace  and  plunged 
into  20  Ibs.  of  water  at  10°  C.,  heated  the  water  to  25°  :  find  the  tempera- 
ture of  the  furnace.     (Specific  heat  as  above.) 

3.  A  mass  of  700  grams  of  copper  at  98°  C.,  put  into  800  grams  of  water 
at  15°  contained  in  a  copper  vessel  weighing  200  grams,  raises  the  tempera- 
ture of  the  water  to  21°  C.  :  find  the  specific  heat  of  copper. 

4.  A  mass  of  200  grams  of  copper,  whose  specific  heat  is  0*095,  is 
heated  to  100°  C.,  and  placed  in  100  grams  of  alcohol  at  8°  C.  contained 
in  a  copper  calorimeter,  whose  mass  is  25  grams,  and  the  temperature  rises 
to  28-5°  C.  :  find  the  specific  heat  of  alcohol.     (London  Inter.  B.Sc.) 

5.  A   glass  bulb  with  a   uniform  fine   stem    weighs  10  grams  when 
empty,  H7'3  grams  when  the  bulb  only  is  full  of  mercury,  and  1197  grams 
when  a  length  of  10*4  cm.  of  the  stem  is  also  filled  with  mercury  :  calcu- 
late the  relative  coefficient  of  expansion  for  temperature  of  a  liquid  which, 
when  placed  in  the  same  bulb,  expands  through  a  length  of  from  10*4  to 
12*9  cm.  of  the  stem  when  warmed  from  o°  C.  to  28°  C.     The  density 
of  mercury  is  13*6  grams  per  c.cm.     (London  Inter.  B,Sc.) 

6.  The  specific  heat  of  mercury  is  said  to  be  —  :  what  does  this  mean  ? 

If  the  heat  yielded  by  I  kilogram  of  water  in  cooling  down  from  100°  to 
o°C.  were  employed  in  heating  10  kilograms  of  mercury  initially  at  20°,  to 
what  temperature  would  the  mercury  be  raised  ?  (London  Matric.) 

7.  Describe  Bunsen's  calorimeter.      If  100  c.cm.  of  water  in  freezing 
become  109  c.cm.  of  ice,  and  the  introduction  of  20  grams  of  mercury  at 
100°  C.  into  a  Bunsen  calorimeter  causes'  the  end  of  the  column  to  move 
through   74  mm.  in  a  tube  I    square   mm.   in   section,  find  the  specific 
heat  of  mercury.     (The  heat  required  to  melt  I  gram  of  ice  is  80  units.) 
(London  Matric.) 

8.  200  grams  of  water  at  99°  C.  are  mixed  with  200  c  cm.  of  milk  of 
density  i'O3  at  15°  C.  contained  in  a  copper  vessel  of  thermal  capacity 
equal  to  that  of  8  grams  of  water,  and  the  temperature  of^  the  mixture  is 
57°  C.  :   if  all  the  heat  lost  by  the  water  is  gained  by  the  milk  and  the 
copper,  what  is  the  specific  heat  of  milk  ? 

9.  What  is  meant  by  the  specific  heat  of  water  at  20°  C.  ? 

10.  Describe    Lavoisier    and  Laplace's    calorimeter.       What   are   its 
defects?     In  an  experiment,  10  Ibs.  of  a  metal  are  inserted  at  215°  C.,  and 
2\  Ibs.  of  water  run  from  the  apparatus  :  determine  the  specific  heat  of  the 
metal. 

11.  What  is  Dulong  and  Petit's  law?    What  explanation  can  you  give 
of  the  case  of  the  diamond  ?     It  is  found  that  the  specific  heat  of  a  new 
metal  is  0*073  :  now  could  you  approximately  estimate  the  atomic  weight? 

12.  Find  the  amounts  of  heat  required  to  raise  I  cubic  metre  of  air 
and  of  hydrogen  from  o°  to  100°  C. 


103 


CHAPTER  VI. 
CHANGE  OF  STATE— THE  NATURE  OF  HEAT. 

70.  General. — The  change  in  molecular  state  can  be  illus- 
trated from  the  forms  of  water.  If  pieces  of  ice  weighing 
i  Ib.  be  placed  in  a  beaker  in  which  is  inserted  a  thermo- 
meter, and  the  beaker  be  plunged  into  a  mixture  of  ice  and 
salt,  the  thermometer  will  soon  indicate  a  temperature  below 
zero.  Let  us  suppose  that  it  indicates  —  6°  C. 

(1)  Remove  the  beaker  and  let  it  stand  in  the  room.     Stir 
the  ice  during   the  process  in  order  to  keep  it  at  the  same 
temperature  throughout  its  mass.     Heat  is  absorbed  from  the 
room,  and  the  temperature  rises  steadily  to  o°  C.     If  the  mean 
specific  heat  of  ice  be  0-504  between   —6°  and  o°,  then  0-504 
X  6  =  3 '024  thermal  units  are  absorbed. 

(2)  The  ice  begins  to  melt,  and  the  temperature  remains  at 
o°  until  the  whole  is  changed  into  liquid  water.     The  mass  has 
continued  to  absorb  heat,  but  the  heat  has  not  affected  the 
temperature  ;•  it  has  been  used  in  changing  the  pound  of  ice  at 
o°  to  water  at  o°.     We  may  state  for  the  present  that  this  will 
take  80  thermal  units,  the  unit  being  the  heat  required  to  raise 
i  Ib.  of  water  through  i°  C. 

(3)  As  soon  as  the  melting  is  completed,  the  temperature 
again  begins  to  rise  until  it  reaches  that  of  the  temperature 
of  the   room.      Place  the  beaker   now  over  a  flame.      The 
increase   in    temperature    continues    until   the    thermometer 
indicates  100°  C.      In  this  stage  the  water  has  absorbed   100 
units  of  heat,  if  we  take  i  as  the  mean  specific  heat  of  water. 

(4)  The  liquid  changes  into  water  vapour,  and  during  the 


104  Heat 

process  the  temperature  remains  steady  until  the  whole  has 
been  changed.  The  absorption  of  heat  has  gone  on,  and,  as 
will  be  seen,  537  units  have  been  absorbed.  The  volume 
at  100°  under  normal  pressure  is  1624  times  the  volume  of  i  Ib. 
of  water  at  100°. 

(5)  If  the  vapour,  by  any  arrangement,  were  kept  in  a 
closed  vessel,  the  temperature  would  again  rise.  If  0*48  be 
the  mean  specific  heat  of  water  vapour,  and  200°  C.  the  final 
temperature,  it  would  absorb  0*48(200  —  100)  =  48  thermal  units. 

Further  changes  would  ultimately  take  place  if  the  tem- 
perature were  high  enough.  The  water  would  disassociate 
into  oxygen  and  hydrogen.  This  effect  we  shall  not  discuss. 

If  the  process  be  reversed,  we  should  have  — 

(5)  The  cooling  of  the  vapour  from  2co°  to  100°,  and  48 
units  of  heat  would  be  liberated. 

(4)  The  changing  of  vapour  at  100°  into  water  at  100°,  and 
the  liberation  of  537  units  of  heat 

(3)  The  cooling  of  water  from  100°  to  o°,  and  the  liberation 
of  100  units. 

(2)  The  changing  of  water  at  o°  into  ice  at  o°,  and  the 
liberation  of  80  units. 

(i)  The  cooling  of  ice  at  o°  to  ice  at  —6°,  and  the  libera- 
tion of  3 '024  units  of  heat. 

Similar  illustrations  could  be  obtained  from  other  substances, 
although,  with  the  means  at  our  command,  all  the  stages  could 
not  be  completely  illustrated  from  each  substance.  Thus  iron 
would  only  show  (i),  (2),  and  part  of  (3) ;  alcohol,  (2),  (3), 
(4),  and  (5);  oxygen,  (3),  (4),  and  (5). 

Attending  more  particularly  to  the  stages  (2)  and  (4),  called 
respectively  those  of  fusion  and  vaporization — or,  when  the  pro- 
cesses are  performed  in  the  reverse  way,  solidification  and  lique- 
faction— we  have  to  observe  that,  if  the  experiments  be  repeated 
with  pure  water  under  normal  atmospheric  pressure,  the  tem- 
perature of  fusion  (or  melting  point)  and  the  temperature  of 
solidification  is  always  at  o°  C.,  and  that  the  temperature  of 
vaporization  (boiling  point),  or  the  temperature  of  liquefaction 
under  similar  conditions,  is  always  at  TOO°  C. ;  and  further, 
that  the  number  of  units  of  heat  necessary  to  complete  the 


Change  of  State — The  Nature  of  Heat          105 

process  of  fusion  (or  the  number  of  units  of  heat  liberated 
in  solidification)  is  constant,  and,  for  the  present,  may  be 
taken  as  80  units;  also  that  the  number  of  units  of  heat 
necessary  to  completely  effect  the  vaporization  (or  the  number 
liberated  during  liquefaction)  is  constant,  and  may  be  taken 
as  equal  to  537. 

71.  The  Nature  of  Heat.—  Caloric. — Heat  has  been 
spoken  of  as  entering  a  body  and  raising  its  temperature ;   of 
leaving  it  and  lowering  its  temperature  ;  and  certain  effects  of 
expansion,  of  rise  of  temperature,  and  of  change  of  state  from 
solid  to  liquid,  from  liquid  to  gas,  have  been  observed.     The 
early  idea  was   that  heat  was  an   imponderable   fluid  called 
caloric,  which  entered  a  body  and  raised  its  temperature,  and 
which,  on  leaving  it,  lowered  the  temperature ;   or  entering  the 
body  during  change  of  state — as  in  melting   ice  at  o°  C.  to 
water  at  o°  C. — the  heat  became   "  latent,"    this  latent  heat 
becoming  evident  to  the  senses  when  the  water  changed  into 
ice.     Water  at  o°  was,  according  to  this  theory,  ice  at  o°  -h  a 
certain  amount  of  caloric. 

Heat  obtained  by  friction  was  explained  by  stating  that, 
in  rubbing,  part  of  the  caloric  was  squeezed  out  of  the  bodies. 

Some  of  the  supporters  of  the  theory  even  experimented  with 
a  view  to  discover  the  change  in  weight  when  caloric  entered 
or  left  a  substance,  and  came  to  the  conclusion  that  the  ad- 
mission of  caloric  made  a  substance  lighter.  The  differences 
were  shown,  however,  to  be  due  to  experimental  errors.  La- 
voisier showed  conclusively  that  a  flask  of  water  hermetically 
sealed  showed  no  change  of  weight  when  the  water  was  frozen. 
He  further  enclosed  six  grains  of  phosphorus  in  a  strong  flask, 
ignited  it  by  the  sun's  rays,  and  proved  that,  after  combustion, 
there  was  no  change  in  weight.  This  did  not,  however,  dis- 
prove the  idea  of  an  imponderable  fluid,  and  the  entrance  of 
this  fluid  into  the  spaces  between  the  particles  of  a  substance 
gave  an  apparently  satisfactory  view  of  increase  in  bulk  in 
expansion,  fusion,  and  vaporization. 

72.  Rumford's  and  Davy's  Experiments. — Rumford, 
in   observing   the   boring   of  cannon,  noticed   the  enormous 
amount  of  heat  generated,  and  performed  a  series  of  experi- 


106  Heat 

ments  to  determine  from  whence  the  caloric  was  derived.  He 
first  examined  the  case  according  to  the  then  theory ;  that  is, 
he  collected  the  fine  borings  and  chips,  heated  them  to  the 
boiling  point  of  water,  placed  them  in  a  simple  calorimeter, 
and  noted  the  rise  in  temperature.  He  repeated  the  experi- 
ment with  slices  of  equal  mass  cut  from  the  solid  metal,  and 
obtained  the  same  rise  in  temperature.  He  concluded  that 
the  heat  generated  in  boring  was  not,  therefore,  due  to  any 
latent  heat  squeezed  out  of  the  metal.  To  satisfy  one  hypo- 
thesis of  the  caloric  theory,  the  capacity  for  heat  of  the  borings 
should  have  been  less  than  the  capacity  for  heat  of  the  solid 
metal.  His  reasoning  was  not  quite  conclusive,  as  the  change 
in  the  state  of  the  filings  and  borings,  compared  with  the 
molecular  condition  of  the  solid  slices,  might  have  affected 
the  result.  Later  examination  showed,  however,  that  this  did 
not  do  so.  (See  end  of  §  68.) 

In  another  experiment  he  arranged  so  that  the  borer 
and  block  of  metal  were  surrounded  with  water.  He  fixed 
a  cylinder  of  brass,  partly  bored,  in  a  box  containing  18  Ibs. 
of  water.  The  borer  revolved  32  times  in  a  minute,  and  was 
moved  by  machinery.  The  initial  temperature  was  60°  F. ; 
in  an  hour  it  was  107°  F. ;  and  in  2^  hours  it  boiled.  He 
estimated  the  heat  produced  as  being  equal  to  nine  wax  candles 
each  three  quarters  of  an  inch  in  diameter,  burning  the  same 
length  of  time. 

The  source  of  heat  could  not  be  from  change  of  capacity, 
as  shown  above ;  it  could  not  come  from  the  air,  as  this  was 
excluded  from  the  apparatus;  nor  from  the  water,  since  it 
underwent  no  chemical  change ;  nor  from  surrounding  objects, 
seeing  that  they  took  heat  from  the  metal.  He  says — 

"The  result  of  this  experiment  was  very  striking,  and  the 
pleasure  it  afforded  amply  repaid  me  for  all  the  trouble  I  had 
taken  in  contriving  and  arranging  the  complicated  machinery 
used  in  making  it.  The  cylinder  had  been  in  motion  but  a 
short  time  when  I  perceived,  by  putting  my  hand  into  the 
water  and  touching  the  outside  of  the  cylinder,  that  heat  was 
generated." 

Rumford's  conclusion  was,  "  In  reasoning  on  this  subject 


Change  of  State — The  Nature  of  Heat          107 

we  must  not  forget  that  most  remarkable  circumstance,  that 
the  source  of  the  heat  generated  by  friction  in  these  experi- 
ments appeared  to  be  inexhaustible.  It  is  hardly  necessary  to 
add  that  that  which  any  isolated  body  or  system  of  bodies  can 
continue  to  furnish  without  limitation,  cannot  possibly  be  a 
material  substance ;  and  it  appears  to  me  extremely  difficult, 
if  not  quite  impossible,  to  form  any  distinct  idea  of  anything 
capable  of  being  executed  and  communicated  in  these  experi- 
ments except  motion." 

Rumford's  reasoning  amounts  to  this — 

(1)  The  calorists  say  that  heat  is  an  imponderable  sub- 
stance. 

(2)  They  have  an  additional  hypothesis,  namely,  that  the 
capacity  of  a  body  for  heat  depends  upon  the  quantity  of 
heat  it  already  contains. 

(3)  The  experiments    show  that   the  capacities  for  heat 
of  the  borings  and  the  solid  are  equal. 

(4)  Therefore  the  theory  of  caloric  is  untrue.  . 

But  the  logical  conclusion  is  that  the  hypothesis  in  (2) 
is  untrue,  and  therefore,  as  Sir  William  Thomson  points  out, 
if  calorists  gave  up  this  hypothesis,  Rumford's  experiments 
had  not  disproved  that  heat  is  an  imponderable  substance. 

Rumford  should  have  reduced  both  the  solid  iron  and 
the  borings  to  the  same  molecular  condition.  He  might,  for 
example,  have  melted  both,  and  compared  the  quantity  of 
heat  necessary  to  melt  each ;  or  he  might  have  dissolved 
them  in  hydrochloric  acid,  and  obtained  a  solution  of  iron 
chloride,  and  compared  the  amount  of  heat  generated  in  each 
process;  the  differences  by  either  method  would  have  been 
inappreciable.  He  then  could  have  fairly  concluded  that 
there  was  nothing  in  the  condition  of  the  metal  in  the  two 
states  to  account  for  the  quantity  of  heat  generated  during 
the  abrasion  of  the  metals,  and  that,  therefore,  heat  was  not 
a  material  substance. 

Rumford  made  a  determination  of  the  amount  of  work 
required  to  raise  i  Ib.  of  water  through  i°.  The  work,  he  said, 
could  be  done  by  one  horse  in  a  given  time,  and  the  results 
do  not  greatly  differ  from  later  determinations. 


io8  Heat 

Boyle  had  proved  that  heat  was  produced  by  rubbing 
metals  in  vacua.  Sir  Humphry  Davy  performed  an  experiment 
to  test  the  caloric  theory,  and  believed  that  he  disproved  it. 

He  rubbed  together  two  pieces  of  ice  until  they  melted, 
taking  precautions  to  prevent  heat  from  any  source  entering 
the  ice.  The  friction  was  performed  by  clockwork  in  vacuo^ 
and  the  complete  apparatus  was  surrounded  by  melting  ice. 
Sufficient  heat  was  produced  by  friction  to  melt  the  ice. 
Davy  then  reasoned  in  this  way :  If  heat  be  an  imponderable 
fluid,  caloric,  then,  in  the  above  experiment,  seeing  that  caloric 
cannot  be  obtained  from  surrounding  bodies  or  from  the  air, 
but  has  been  squeezed  out  of  the  ice,  the  water  formed  should 
have  a  less  capacity  for  heat  than  the  ice  from  which  it  was 
formed.  But  the  capacity  for  heat  of  water  is  well  known  to 
be  greater  than  that  of  ice,  and  ice  must  have  an  absolute 
quantity  of  heat  added  to  it  before  it  can  be  converted  into 
water.  Friction,  consequently,  does  not  diminish  the  capacity 
of  bodies  for  .heat.  The  heat  needed  to  melt  the  ice,  then,  can 
only  be  the  result  of  friction.  (Compare  Rumford's  reasoning.) 

"The  immediate  cause  of  the  phenomenon  of  heat,  then, 
is  motion ;  and  the  laws  of  its  communication  are  precisely 
the  same  as  the  laws  of  the  communication  of  motion." 

The  ordinary  experiments  that  when  a  body  is  rubbed — 
a  button  on  a  piece  of  wood — it  becomes  warm,  or  when  heat 
is  generated  by  hammering,  as,  for  example,  when  we  raise  the 
temperature  of  a  piece  of  lead  by  a  series  of  blows, — can,  it 
should  be  observed,  be  readily  explained  by  the  caloric  theory. 

Speaking  of  experiments  such  as  Davy's,  Dr.  Young  puts 
the  case  against  the  caloric  theory — 

"  If  the  heat  is  neither  received  from  the  surrounding 
bodies,  which  it  cannot  be  without  a  depression  of  their 
temperature  •  nor  derived  from  the  quantity  already  accumu- 
lated in  the  bodies  themselves,  which  it  could  not  be  even  if 
their  capacities  were  diminished  in  any  imaginable  degree;  — 
there  is  no  alternative  but  to  allow  that  heat  must  be  actually 
generated  by  friction  ;  and  if  it  is  generated  out  of  nothing  it 
cannot  be  matter,  nor  even  an  immaterial  or  semi-material 
substance." 


Change  of  State — The  Nature  of  Heat         109 

The  examination  of  the  relation  between  heat  and  work  was 
continued,  among  others,  by  Mayer  and  Joule. 

73.  Heat  and  Work.— The  experi- 
ments of  Rumford  and  Davy  were  bringing 
into  prominence  the  fact,  not  only  that 
work  produced  heat,  but  that  the  heat 
produced  was  the  result  of  the  work  alone, 
and  that  the  heat  might  be  measured  in 
terms  of  work. 

Simple  experiments  upon  solids  and 
liquids  have  been  mentioned  The  fire- 
syringe  experiment  (Fig.  46)  illustrates  it 
in  the  case  of  gases.  The  piston  works 
lightly  in.  the  glass  barrel  A  piece  of 
cotton  wool  is  moistened  with  carbon 
disulphide,  and  placed  in  a  small  hole 
in  the  base  of  the  piston.  On  pressing 
the  piston  quickly  down,  the  heat  pro- 
duced by  the  compression  of  the  air  is 
sufficient  to  ignite  the  cotton  wool. 

In  a  later  chapter  the  converse  of 
this  experiment  will  be  seen. 

Another  experiment,  due  to  Tyndall, 
is  shown  in  Fig.  47.  A  brass  tube  is  fitted 


FIG.  46. 


FIG.  47. 


no  Heat 

to  the  stand,  and  made  to  revolve  rapidly  by  turning  the  wheel. 
It  is  clasped  tightly  between  the  presser  lined  with  leather. 
Soon  the  friction  causes  the  ether  to  boil  and  the  cork  is 
expelled. 

If  a  piece  of  lead  be  hammered  for  a  few  seconds,  a  dis- 
tinct rise  of  temperature  will  be  detected,  either  by  testing  it 
with  the  differential  thermometer  or,  more  markedly,  with  the 
thermo-pile. 

74.  Note  on  Dynamical  Units. — Every  quantity  is 
expressed  by  a  phrase  that  consists  of  two  parts  :  (i)  the 
numerical  portion,  and  (2)  the  unit  of  the  quantity;  for 
example,  the  distance  14  yards  can  be  expressed  as  14  yards 
(unit  i  yard),  7  fathoms  (unit  i  fathom),  42  feet  (unit  i 
foot),  etc. 

Dynamical  quantities  can  all  be  derived  from  units  of 
length,  mass,  and  time ;  these  are  called  fundamental  units. 

The  standard  of  length  in  England  is  i  yard ;  that  of 
France,  i  metre.  Both  standards  are  preserved  with  care,  and 
copies  have  to  be  made  at  definite  temperatures  (§  22).  The 
usual  practical  and  scientific  units  used  are  sub-multiples  of 
these :  i  foot  and  i  cm. 

i  metre  =  39*37  inches;  and  therefore  i  foot  =  30-48  cm. 

The  symbol  for  length  is  s  (space). 

The  English  standard  of  mass  (m)  is  i  Ib. ;  the  French 
standard  is  i  kilogram.  Each  is  a  piece  of  platinum  pre- 
served in  the  respective  countries.  The  French  scientific  unit 
is  i  gram. 

i  Ib.  =  0-45359  kilogram  =  4S3'59  grams 

The  general  unit  of  time  (/)  is  i  second. 

The  system  of  units  derived  from  the  centimetre,  the  gram, 
and  the  second  is  called  the  centimetre-gram-second  system, 
written  C.G.S.  system.  The  English  adopt  a  foot-pound- 
second  system ;  and  a  system  could  be  built  on  any  selected 
units  of  length,  mass,  and  time. 

From  the  units  of  length,  mass,  and  time  are  derived  the 
units  of — 


Change  of  State — The  Nature  of  Heat         in 

Area :  i  square  foot ;  r  square  cm. 

Volume  :  i  cubic  foot ;  i  c.cm. 

Other  units,  such  as  i  square  inch,  i  square  metre,  are  in 
use ;  but  we  are  dealing  more  particularly  with  those  needed 
for  our  purpose. 

i  c.cm.  of  water  at  4°  C.  weighs  i  gram 

i  litre  =  1000  c.cm. 

i  cubic  foot  of  water  weighs  1000  ozs.  nearly 

i  gallon  =  277*276  cubic  inches 

i  gallon  of  water  weighs  10  Ibs.  nearly 

Speed  (v)  is  the  rate  at  which  a  body  moves  with  respect 
to  surrounding  objects.  Sometimes  the  term  "velocity"  is 
used  instead  of  "  speed."  In  speed  motion  alone  is  considered ; 
velocity  implies  the  further  consideration  of  direction. 

Speed,  if  uniform,  is  measured  by  the  number  of  units  of 
length  (feet  or  cm.)  passed  over  in  unit  time  (i  second). 

The  units  of  speed  are:  (i)  English,  i  foot  per  second; 
(2)  C.G.S.,  i  cm.  per  second. 

i  foot  per  second  =  30-48  cm.  per  second 

Acceleration  (a)  is  the  rate  at  which  a  body  changes  its 
speed;  for  example,  at  one  instant  the  speed  of  a  body  is 
40  feet  per  second,  a  minute  later  it  is  56  feet  per  second,  a 
minute  later  72  feet  per  second.  If  the  increase  has  been 
uniform,  the  speed  has  changed  at  the  rate  of  16  feet  per 
second  in  every  minute.  Time  is  repeated  twice,  and  it  is 
usual  to  keep  the  second  as  unit  in  each  case. 

The  units  of  acceleration  are :  (i)  English,  i  foot  per 
second  per  second ;  (2)  C.G.S.,  i  cm.  per  second  per  second 

i  foot  per  second  per  second  =  30*48  cm.  per  second  per  second 

If  we  attempt  to  move  a  body  at  rest  we  are  conscious  of 
muscular  effort,  whether  we  succeed  in  starting  it  in  motion 
or  not.  There  is  a  similar  muscular  effort  exerted  when  we 
attempt  to  stop  or  retard  the  motion  of  a  moving  body.  In 
both  cases  we  exert  force. 

Newton's  first  law  of  motion  states — 

That  every  body  continues  in  a  state  of  rest  or  of  uniform 


H2  Heat 

motion  in  a  straight  line  except  in  so  far  as  it  is  compelled  to 
alter  that  state  by  some  force. 

Momentum  is  the  product  of  mass  and  velocity  (mv).  The 
units  of  momentum  are  :  (i)  the  momentum  of  a  mass  of 
i  Ib.  moving  with  a  velocity  of  i  foot  per  second ;  and  (2)  the 
momentum  of  a  mass  of  i  gram  moving  with  a  velocity  of 
i  cm.  per  second. 

Newton's  second  law  is — 

Change  of  momentum  is  proportional  to  the  force  acting  upon 
a  body,  and  takes  place  in  the  direction  in  which  the  force  acts. 
The  change  can,  of  course,  only  be  effected  in  a  certain  time. 

A  mass  of  6  Ibs.  moves  with  a  speed  of  8  feet  per  second ; 
a  second  later  its  speed  is  3  feet  per  second. 

Momentum  at  beginning  of  second  =  6  x  8  =  48 
„  „          end        „       „       =6x3  =  18 

.*.  force  acting  for  i  second  against  the  body  is  proportional  to  30 
Or  force  X  time  =//  =  m(v'  -  v") 

The  change  in  velocity,  if  uniform,  divided  by  the  time,  is 
the  acceleration ;  or  it  is  the  acceleration  at  the  particular  time, 
if  we  make  the  time  small  enough. 

,         it  -  tf 
.*.  /  =  m  — T-  -  =  ma 

The  unit  of  force  is  that  force  which,  acting  on  unit  mass 
for  unit  time,  produces  unit  change  of  momentum. 

(1)  English  :  that    force  which,  acting  on  the  mass  of  i  Ib., 
changes  its  velocity  i  foot    per  second  in  every  second.     This 
unit  is  called  fa&  pounded. 

(2)  The  C.G.S.  unit,  called  the  dyne,  is  that  force  which, 
acting  on  the  mass   of  i  gram,  produces  a  change  of  velocity 
of  i  cm.  per  second  in  every  second. 

In  the  above  example  the  force  is  30  poundals. 

i  pndl.  actg.  on  i  Ib.  prod.  chng.  of  vel.  of  i  ft.  per  sec.  in  ev.  sec 
•M          >,          453'59gr-     »  »        30-48  cm.     „ 

.M          „  igr.          „  „      453*59  X  30'48  cm.  „ 

.M          „  igr.          „  ,,13,825-4001.       „         „ 

/.  T  poundal  =  13,825-4  dynes 


Change  of  State — The  Nature  of  Heat         113 

i  Ib.  has  been  used  as  the  unit  of  mass ;  it  is  sometimes 
taken  as  a  unit  of  weight.  Either  idea  may  be  retained  if  we 
are  careful  to  keep  to  the  same  name  throughout.  When  we 
take  i  Ib.  as  unit  of  mass,  we  consider  merely  the  quantity  of 
matter  it  contains — a  quantity  that  can  be  used  for  manu- 
facturing or  other  purposes.  As  unit  of  weight,  we  are  con- 
sidering its  push  downwards,  or  the  pull  it  exerts  on  a  spring- 
balance,  for  example  ;  that  is,  we  consider  it  as  a  force.  That 
it  is  best  to  consider  it  as  a  unit  of  mass  will  be  seen  from  the 
following  applications. 

If  the  standard  pound  be  suspended  from  a  very  delicate 
spring-balance  that  indicates  a  pull  of  i  Ib.  on  the  scale  in 
London,  then,  if  we  move  to  the  equator,  it  will  register  less 
than  i  Ib. ;  that  is,  the  force  downwards,  due  to  gravitation,  has 
decreased.  Nearer  to  the  pole  it  will  indicate  more  than  i  Ib. 
The  pull  downwards,  then,  is  not  constant,  although  there  has 
been  no  change  in  the  mass  ;  i  Ib.  of  iron  will  make  into  the 
same  number  of  the  same  kind  of  nails  in  any  of  the  three  places. 

If  the  i  Ib.  be  allowed  to  fall  from  rest  at  Greenwich,  its 
velocity,  as  shown  by  experiment,  at  the  end  of  the  first  second 
will  be  32*2  feet  per  second. 

Momentum  at  starting  =  i  X  o  =  o 

„  gained  in  i  second  =  i  x  32*2  =  32-2  units 

The  force  acting  upon  it,  due  to  gravity,  is  the  weight  of 
i  Ib.  This  force,  acting  on  i  unit  of  mass,  produces  a  change 
of  32-2  units  of  momentum  per  second  in  i  second  ;  that  is, 
the  weight  of  i  Ib.  at  Greenwich  =  32*2  poundals.  At  Paris 
the  velocity  at  the  end  of  the  first  second  would  be  32-18  feet 
per  second. 

.*.  the  weight  of  r  Ib.  at  Paris  =  32-18  poundals 

Similarly,  the  weight  of  i  Ib.  at  the  equator  =  32-09  poundals. 

The  weight  of  i  Ib.  is  called  the  force  of  i  Ib. 

The  differences  in  the  values  of  the  weight  of  i  Ib.  are 

small.     Between  London  and  Paris  it  is  less  than of  the 

4000 

force  at  either  place,  and  can  be  neglected  in  engineering  and 

i 


114  Heat 

practical  operations,  i  Ib.  considered  as  a  force  is  commonly 
used  by  practical  men  because  it  is  a  more  suitable  unit 
than  i  poundal,  which  is  roughly  equal  to  i  Ib.  force  4- 3 2  = 
the  weight  of  £  oz.,  and  is  therefore  inconveniently  small. 

If  we  take  g  to  express  the  acceleration  due  to  gravity, 
which  varies  with  the  latitude  and  distance  above  the  sea-level, 
then  always — 

i  Ib.  weight  =  g  poundals 

i  gram  weight  =  g  (in  centimetres)  dynes 

The  acceleration  due  to  gravity,  in  centimetres  per  second, 
is  978*1  at  the  equator,  980*94  at  Paris,  981*17  at  Greenwich, 
and  98 1 -34  at  Manchester. 

A  force  does  work  (w)  when  it  moves  its  point  of  applica- 
tion. The  measure  of  the  work  is  the  product  of  the  force 
and  the  distance  moved  in  the  direction  in  which  the  force 
acts. 

w  —  fs 

(1)  The  English  unit  is  the  foot-poundal  \  it  is  the  work 
done  by  i  poundal  in  moving  through  i  foot. 

(2)  The  C.G.S.  unit,  the  erg,  is  the  work  done  by  i  dyne 
in  moving  over  i  cm. 

i  foot-poundal  =  13,825  dynes  working  through  30*48  cm.(i  foot) 
=  13,825  x  30*48  dynes      „        i  cm. 
=  421,394  ergs 
=  4-214  x  io5  ergs  nearly 

The  common  units,  based  on  the  weight  of  i  Ib.,  i  gram, 
i  kilogram,  etc.,  are— r 

(1)  English  :  work  done  by  a  force  of  i  Ib.  working  through 
i  foot,  called  a  foot-pound. 

i  foot-pound  =  g foot-poundals  =  32*2  x  4*214  X  io5  ergs 
=  1*36  x  i  o7  ergs  nearly  (at  Greenwich) 

(2)  French:  i  gram-centimetre  =  .Jfergs  =  981  ergs  nearly 

i  kilogram-metre    ==  (1000  x  100)  gram-cm, 
=  g  X  io5  ergs  =  9*81  x  io7  ergs 

The  poundal  and  foot-poundal,  the  dyne  and  the  erg,  are  in 
all  cases  independent  of  g,  and  are  called  absolute  units. 


Change  of  State  —  The  Nature  of  Heat         115 

The  foot-pound,  the  kilogram-metre,  etc.,  involve  ultimately 
the  value  of  £•,  and  are  called  gravitation  units. 

Energy  is  the  capacity  for  doing  work,  and  is  measured  in 
the  same  units  as  work. 

If  a  mass  of  10  Ibs.  be  raised  6  feet  from  the  ground,  60 
foot-pounds  of  work  are  done.  The  mass,  by  virtue  of  its 
position  in  relation  to  the  earth,  can  in  falling  do  60  foot- 
pounds of  work.  It  can  raise,  with  suitable  machinery,  10  Ibs. 
through  6  feet,  or  20  Ibs.  through  3  feet,  neglecting  friction. 
The  catapult,  when  stretched,  can  do  work  :  it  can  project,  let 
us  say,  a  stone  weighing  i  oz.  to  a  height  of  50  feet,  and 
therefore  can  do  50  -f-  16  =  3!-  foot-pounds,  or  nearly  100  foot- 
poundals,  of  work.  Its  capacity  for  work  depends  upon  the 
relative  positions  of  the  particles  of  the  indiarubber.  The  work 
that  can  be  done  by  a  bent  spring,  by  a  head  of  water,  are 
other  examples  of  capacity  for  work,  due  to  the  positions  of  the 
relative  parts  of  a  system.  This  is  called  potential  energy.  Its 
measure  is  force  into  effective  distance. 

A  body  has  also  a  capacity  for  work  due  to  its  motion. 
The  work  done  by  a  cannon-ball,  the  work  done  (destructive) 
by  ships  in  collision,  the  work  done  by  the  moving  hammer, 
are  examples.  Energy  due  to  motion  is  called  kinetic  energy. 

Kinetic  energy  is  measured  in  absolute  units  by  the  pro- 
duct of  half  the  mass  into  the  square  of  the  velocity. 

A  mass  of  5  Ibs.  is  moving  with  a  velocity  of  10  feet  per 
second. 

T  V   "*  V  T  O  " 

Kinetic  energy  in   foot-poundals  =  -  -  =  250  foot- 


poundals  -  -      =  7||  foot-pounds  (it>  =  32) 

This  is  the  work  the  body  can  do  before  it  is  brought  to 
rest. 

A  mass  of  2  Ibs.,  64  feet  from  the  ground,  possesses 
potential  energy.  It  can,  by  virtue  of  its  position  relative  to 
the  earth,  do  128  foot-pounds,  or  128^  foot-poundals,  of  work. 
If  the  body  falls  freely,  at  the  end  of  i  second  it  will  have 
fallen  approximately  16  feet,  and  will  have  a  velocity  of  32 
feet  per  second. 


1 16  Heat 

(a)  Potential  energy  =  2  x  48  =  96  foot-pounds 

(b)  Kinetic  energy     = =32  foot-pounds 

2  x  g 

Total  energy  =  a  -f-  b  -  128  footpounds 

Just  before  it  touches  the  ground  its  velocity  is  64  feet  per 
second. 

Potential  energy  =  o 

Kinetic  energy     =  —  —  foot-pounds  =128  foot-pounds 

2  X  g 

The  total  energy  of  a  system,  unless  it  acts  upon  or  is  acted 
upon  by  another  system,  is  a  fixed  quantity. 

When  the  mass  strikes  the  ground,  the  energy  is  not 
apparently  as  easy  to  trace.  A  thud  is  heard,  there  is  the 
rebounding  of  the  mass  and  of  particles  of  earth,  but  soon 
all  are  at  rest,  and  energy  seems  destroyed  ;  for  the  energy 
needed  to  start  the  sound-wave  (particles  of  air  must  be  set 
in  motion)  is  small  compared  with  128  foot-pounds.  A  wave 
will  have  been  transmitted  through  the  earth;  this  will  take 
up  some  of  the  energy;  these  waves  die  away,  but  a  certain 
amount  of  heat  is  produced.  The  increase  in  temperature  is 
very  small,  and  soon  there  is  a  'cooling  down,  and  uniformity 
of  temperature  ensues.  The  mass  after  impact  would  be 
found  to  be  heated ;  it  again  soon  cools  down  to  the  surround- 
ing temperature ;  the  energy  has  disappeared,  and  heat  seems 
to  be  the  total  result. 

The  following  sections  will  show  that,  by  doing  work,  heat 
may  be  generated,  and  further,  that  the  amount  of  heat 
generated  is  always  proportional  to  the  work  done  ;  also  that, 
by  the  action  of  heat,  work  can  be  done,  and  the  work  done  is 
proportional  to  the  amount  of  heat  that  disappears.  This 
has  led  to  the  idea  that  heat  is  a  form  of  energy.  It  has 
already  been  stated  that,  in  a  hot  body,  the  particles  are 
vibrating ;  we  may  therefore  consider  sensible  heat  as  kinetic 
energy. 

In  the  illustration  with  the  falling  body,  if  the  total  heat- 
energy  could  be  estimated  when  all  was  apparently  at  rest, 
it  would  equal  128  foot-pounds. 


Change  of  State — The  Nature  of  Heat         117 

From  consideration  of  the  various  forms  of  energy,  the 
doctrine  of  the  conservation  of  energy  has  been  stated  ;  it  has 
been  found  to  answer  every  possible  test,  and  is  one  of  the 
most  important  generalizations  in  science. 

"  The  total  energy  of  any  body  or  system  of  bodies  is  a 
quantity  which  can  neither  be  increased  nor  diminished  by  any 
mutual  action  of  these  bodies,  though  it  may  be  transformed 
into  any  of  the  forms  of  which  energy  is  susceptible." 

75.  Change  of  Units.— Dimensions.— Some  examples 
have  already  been  given.  As  the  original  memoirs  relating  to 
heat  are  in  varying  units,  it  will  be  well  to  consider  the  general 
principles. 

A  speed,  by  definition,  will  always  be  represented  by  a 
length,  L,  divided  by  a  time,  T.  A  speed  of  28  feet  per 

second  =28  second  ;  if  we  wish  to  change  this  into  cm.  per 

minute,  we  only  need  deal  with  the  (— — — rj  unit,  and  then 
multiply  by  the  numerical  portion. 

i  foot  =  30-48  cm. ;  i  second  =  ^-  minute. 

60 

foot     _  ^_ 

d      ^j-  minute 

cm. 
minute  =  1826-8  ^^ 

to   change  the   numerical   value   of  a  speed  in  feet    per 
second  into  cm.  per  minute,  multiply  by  1826-8 


is  called  the  dimensional  equation  of  a  speed.  Length  is 
involved  to  the  first  power  in  the  numerator ;  time,  to  the  first 
power  in  the  denominator.  It  is  sometimes  written — 

V  =  LT-1 

Acceleration  is  a  velocity  -f-  time. 

•      A    --   —   =   —    *    T   —   —     •    r»r    A    —   T.T-2 
. .   A  -   rp         ,j,  -7-  A    —  ,p2  ,    Or  A  — 


foot         30-48  cm. 
i  foot  per  second  =  — r-H  =  V^-  -  =  1826-8  cm.  per 


Ii8  Heat 

An  acceleration  of  100  miles  per  minute  per  minute  = 

mile 

100  X  minute  x  minute  (umts  :  '  ralle'  '  mlnute)'  To  chanSe 
these  to  feet  per  second  per  second,  we  have  length  involved 
to  the  first  power  ;  time,  inversely  as  the  second  power. 

_  mile  _  _  L_         5280  _  foot  _ 

minute  x  minute  ~~  T*  ~~  60  x  60  second  X  second 

44  foot 

~  30      second  X  second 

.*.  100  miles  per  minute  per  minute 

=  —  X  ioo  feet  per  second  per  second 
=  146*6  feet  per  second  per  second 

We.  can,  with  a  greater  expenditure  of  work,  obtain  the 
result  as  follows  :  — 

ioo  miles  per  minute  per  minute 

=  ioo  x  5280  feet  per  minute  per  minute 

ioo  x  5280  , 

=  -  7-^  —  feet   „    second   „   minute 

ioo  x  5280  . 

=  —7  --  ^—  feet    „    second   „   second 

DO  X   OO 

Area  =  L2 
Volume  =  L3 

L      ML 

Momentum  (/x,)  =  mass  x  velocity  =  M  •  ?p  =  ~^- 

u,      ML 
Force  (P)  =  ^  =  -^r 

ML 

Or  F  =  mass  x  acceleration  =  MA  = 


Poundal  units  are  i  lb.,  i  foot,  i  second. 
Dyne          ,,      ,,    i  gram,  i  cm.,  i  second. 

/.   to  change   poundals  into  dynes   (i   lb.  =  453*59   grams  ; 
i   foot  =  30-48   cm.  ;    time   is   unchanged),  multiply  by 

453-59  X  3°-48  ^^^ 


Change  of  State— T  lie  Nature  of  Heat         119 
To  change  poundals  into  a  unit  of  force  measured  in  yards, 


i         i 
XT 


tons,  and  minutes,  the  multiplier  is =  -7 

60      60 

P        M 

A  pressure  is  a  force  per  unit  area  =  |1  =  jnjia 

Density  is  mass  per  unit  volume  =  VQiume  =  j^ 
Work  =  force  X  distance  =  PL  =  -T=J- 


59  X  (3Q-48)2 
/.  i  foot-poundal  =  ^^-   -—~ri  -  -  ergs  =  4-214  x  io5  ergs 

A  I 


i  foot-pound  =  g  foot-poundals  =  ^X  4*214  x  io5 

=  1*36  x  io7  ergs  (if£-  =  32-2) 
i  gram-cm.  =  ^ergs  =  981  ergs  (if^  =  981  cm.) 

/.  i  foot-pound  =  I3Q8II~=  I3)825  gram-cm. 

Or  direct  — 

Work  =  P  X  L 

/.  i  foot-pound  =  453*59  X  30-48  =  13,825  gram-cm. 
(a)  Kinetic  energy  =  %mv\    (b)  Potential  energy  =  F  x  L. 
.*.  dimensions  of  energy  are  — 


i.e.  in  both  cases  the  same  dimensions  as  work. 
Power  is  the  rate  at  which  a  body  works 


=    T        T3 

i  horse-power  =  33,000  foot-pounds  per  minute  =550  foot- 
pounds per  second 
=  550  X  32-2  foot-poundals  per  second 


120  Heat 

55°  X  32-2  X  453'59  X 


=  7-5  X  i  o9  ergs  per  second 

7  '5  X  io9 
=     J  8  -  gram-cm,  per  second 

=  7'6  x  io6  gram-cm,  per  second 
=  76  kilogram-metres  per  second 

In  heat  we  have  a  special  unit  to  introduce  ;  we  must 
have  some  unit  to  express  a  degree  of  temperature  on  any 
scale  :  let  this  be  A. 

The  thermal  unit  is  the  heat  required  to  raise  unit  mass 
of  a  substance  through  i°. 

.*.  dimensions  of  thermal  unit  =  MA 

The  unit  is  sometimes  defined  as  the  quantity  required  to 
raise  unit  volume  through  one  degree  of  its  temperature  ;  its 
dimensions  will  therefore  be  L3A  when  so  denned. 

change  of  length,  area,  or  volume 
The  coefficient  of  expansion  =  ™^al  length,  area,  or  volume 

per  degree  ;  the  fractional  part  is  a  ratio,  and  therefore  has  no 
dimensions  ;  therefore  in  dimensions  —  • 

Coefficient  of  expansion  =  -r 

Specific  heat  is  a  ratio  merely,  and  has  no  dimensions. 
Thermal  capacity  of  unit  volume  is  specific  heat  x  density  ; 

therefore  its  dimensions  are  those  of  density  =  ^ 

It  will  be  shown  in  the  next  section  that  the  unit  of  heat 
=  J  X  units  of  work. 

J,  called  Joule's  equivalent,  is  the  mechanical  equivalent 
of  unit  heat. 

J  is  the  number  of  units  of  work  per  unit  of  heat. 

/.  for  its  dimensions  we   have        2    ~  MA  =  -       •  (work  in 


absolute  units) 

L 

Or  ML  -7-  MA  =  —  (work  in  gravitation  units) 


Change  of  State — The  Nature  of  Heat        121 

/.&,  in  changing  the  value  of  J  from  one  system  of  units  to 
another  system,  the  unit  of  mass  is  never  involved. 

76.  Joule's  Experiments. — One  of  Joule's  methods  was 
practically  that  followed  by  Rumford  (Fig.  48). 

A  copper  vessel,  B,  was  provided  with  a  brass  paddle-wheel 
(indicated  by  the  dotted  lines),  which  rotated  about  a  vertical 


FIG.  48. 

axis,  A.  The  axis  was  rotated  by  the  weights  E  and  F,  the  cord 
for  each  weight  being  so  arranged  that,  falling,  each  weight 
rotated  the  axis  in  the  same  direction.  Each  weight  could  fall 
in  the  experiment  63  feet,  the  distance  being  measured  on  the 
scales  G  and  H. 

The  paddles,  churning  the  water,  raised  its  temperature; 
when  the  weights  had  fallen  to  the  bottom,  by  loosing  a  pin 
(not  shown  in  the  figure)  A  was  disconnected  from  the  paddles, 
and  by  turning  the  handle  at  the  top  of  A,  the  weights  were 
raised,  the  pin  inserted,  and  the  weights  again  allowed  to 
fall. 

The  axes  of  the  pulleys  (D,  C)  were  rested  in  friction- 
wheels,  to  diminish  the  friction  as  far  as  possible.  Small 
corrections  were  made  for  loss  of  heat,  the  heating  of  the 
calorimeter  and  paddle,  radiation,  etc. 

If  m  =  the  weight  of  the  two  weights,  h  the  distance  fallen 


122  Heat 

in  one  descent,  and  v  the  velocity  with  which  they  reach  the 
bottom,  then,  during  each  fall  the  work  done,  in  foot-pounds 

—  w  —  mh  —  £ — v* 

o 

mh  =  potential  energy  =  total  energy  at  the  beginning 
fyntf  JL.  g  -  kinetic  energy  =  total  energy  at  the  end 

/.  w  =  loss  of  energy  =  work  done  by  the  paddles 

If  the  weights  fall  n  times,  the  total  work  done,  in  foot- 
pounds, is  nw  =  W. 

Weight  of  water  in  calorimeter  x  degrees  through  which  the 
temperature  had  risen  =  thermal  units  given  to  the  water  (H). 

The  experiment  was  repeated  in  various  forms  with  the 
greatest  care  to  eliminate  errors,  using  other  liquids,  such  as 
mercury,  to  show  that  the  result  was  independent  of  the 
material;  so  that  we  can  accept  the  numbers  obtained  as 
being  almost  free  from  error;  the  general  result  being  that 

TT 

the  ratio  ^  in  all  experiments  equals  a  constant,  J  =  772  foot- 
pounds when  degrees  Fahrenheit  are  used. 

From  a  large  number  of  similar  experiments  Dr.  Joule 
concluded  that  to  raise  i  Ib.  of  water  i°  Fahrenheit  requires 
the  expenditure  of  772  foot-pounds  of  work.  If  the  thermo- 
meter be  marked  in  Centigrade  degrees,  the  expenditure  of 
work  will  be  1389-6  foot-pounds. 

772  foot-pounds  of  work  is  the  mechanical  equivalent  of 
one  thermal  unit  in  the  Fahrenheit  scale  when  i  Ib.  is  the  unit 
of  weight  and  i  foot  the  unit  of  length. 

If  the  measurements  were  only  needed  for  general  pur- 
poses, no  appreciable  error  would  be  made  by  using  this 
number;  but  as  the  measure  of  the  work  done  varies  with 
the  force  due  to  gravity,  and  this  varies  with  the  latitude  and 
the  distance  above  the  sea-level,  the  following  particulars 
should  be  added  :  The  experiment  was  performed  at  Man- 
chester, and  the  thermal  unit  was  the  heat  needed  to  raise  i  Ib, 
of  water  from  50°  to  51°  F. 

This  is  a  result  obtained  from  experiment,  and  does  not 


Change  of  State  —  The  Nature  of  Heat         123 

give  us  any  definite  clue  to  the  nature  of  heat;  nor  is  any 
hypothesis  as  to  the  nature  of  heat  involved.  What  the 
experiments  really  show  is  that  — 

Whenever  a  certain  amount  of  energy  is  converted  into 
heat,  the  number  of  units  of  heat  is  always  proportional  to 
the  energy. 

This  is  called  the  First  Law  of  Thermo-dynamics^  and  is 
expressed  as  follows  :  — 

"  When  equal  quantities  of  mechanical  effect  are  produced 
by  any  means  whatever  from  purely  thermal  sources,  or  lost 
in  purely  thermal  effects,  equal  quantities  of  heat  are  put  out 
of  existence,  or  are  generated.  " 

The  value  of  J  in  feet  and  C.°  is  772  x  -  =  1390  foot-pounds 

i  Ib.  =  0-4536  kilog.  j  i  foot  =  0-3048  metre 

i  th.  unit  (kilog.,  C.°)  =  1390  —  0-4536  foot-pounds 

=  1390  —  0-4536  x  0-3048  metre-pounds 
=  1390  —  0-4536  x  0-3048  x  0-4536  m.-kgs. 
=  424  kilogram-metres  of  work 
/.  J  =  424  when  the  units'  are  i  kilogram,  i  metre,  i°  C. 

We  deduce  this  at  once  by  units  (see  end  of  §  75). 


feet  feet  feet  0-3048  metre 

/.  772  -jn>  =  772-^o  =  T39°  QCT  =  1390  -       Co 

metre  cm. 

=     424       -£-b—      =      42,400      £-Q 

In  Absolute  Units.  —  i  unit  of  heat  =  772^  foot-poundals 
=  772X32  '2  foot-poundals  ~  24,858  foot-poundals  (if  £-=32*2). 

L2 

Now,  the  dimensions  of  J  are    rm 


.'.  24,858  (foot,  Ib.,  second,  system)  =  24,858  •     s.        =  4*X57 
X  io7  (C.G.S.  system) 

The  work  of  Rowland  makes  772  too  low  for  the  value  of 
J;  777  is  probably  nearer  the  true  result.     He  also  obtained. 


1 24  Heat 

numoers  which  indicate  that  J  is  not  constant ;  he  found  the 
minimum  value  at  29°  C.  (774*5).  The  value  772  (4*16  x  io7 
in  C.G.S.)  is,  however,  retained  throughout  this  work. 

77.  Molecular  Theory  of  Heat.— According  to  modern 
views,  all  substances  are  supposed  to  be  made  up  of  molecules, 
a  molecule  being  the  smallest  possible  portion  of  the  substance 
that  can  exist  by  itself  as  that  substance.  These  molecules 
are  so  small  that  they  are  far  beyond  recognition  by  the  most 
powerful  microscope.  Each  molecule  may  consist  of  one,  two, 
or  more  atoms. 

The  molecules  are  ever  in  motion.  They  are  highly 
elastic,  so  that,  if  compressed  vibrations  like  those  of  a  sound- 
ing bell  or  tuning-fork  take  place,  there  may  also  be  move- 
ments among  the  component  parts  of  the  molecules.  The 
molecules  may,  in  addition,  be  rotating,  and  may  also  be  moving 
as  a  whole  along  a  certain  path ;  the  form  and  extent  of  the 
motion  depend  upon  the  particular  condition  of  the  substance. 
If  the  substance  be  a  solid,  then  the  molecules  are  more  or  less 
restrained  in  their  movements ;  they  never  move  far  from 
certain  definite  centres.  They  whirl  round  these  centres  like 
planets  in  their  orbits,  but  never  lose  their  connection  with  the 
adjacent  molecules  unless  subjected. to  some  great  shock.  Solids 
thus  resist  any  change  of  form. 

In  liquids  the  molecules  can  move  from  place  to  place  freely, 
slip  easily  over  each  other,  so  that  the  form  of  a  liquid  readily 
changes.  The  cohesion  between  the  molecules  in  liquids  is,  how- 
ever, great,  so  that  it  is  difficult  to  affect  the  volume  of  the  liquid. 

When  the  liquid  is.  changed  to  a  gas,  in  addition  to  an 
increase  in  the  various  forms  of  movement,  the  molecules 
move  rapidly  among  each  other,  colliding,  moving  in  straight 
lines,  and  impinging  against  the  walls  of  the  containing  vessel 
and  against  each  other.  For  an  average  distance  that  is 
always  much  greater  than  the  diameter  of  the  molecules,  they 
meet  with  no  obstruction.  This  is  called  the  free  mean  path 
(§  I73)*  The  effect  of  the  radiometer  is  obtained  by  making, 
by  exhaustion,  this  free  mean  path  sufficiently  great.  No 
matter  how  great  the  space  into  which  a  portion  of  the  gas 
be  introduced,  the  molecules  separate  and  attempt  to  penetrate 


Change  of  State— The  Nature  of  Heat         125 

into  every  part  As  long  as  there  is  an  envelope  they  will 
press  against  it.  They  offer  as  a  mass  comparatively  slight 
resistance  to  change  of  form  or  of  volume  under  ordinary 
conditions,  and  if  subjected  to  force  rapidly  close  up. 

78.  Internal  and  External  Work. — The  general  effect 
of  heat  upon  a  body  is  to  increase  the  intensity  of  the  motion 
of  the  molecules ;  and  it  may  also  separate  them  further  from 
each  other,  moving  the  sides  of  the  containing  vessel  and 
overcoming  resistance.  Thus  we  may  have  an  internal  effect 
and  an  external  effect. 

(1)  If  the  substance,  on  being  heated,  increases  in  volume, 
then  it  must  push  back  the  surrounding  substances — air,  for 
example;   in   so   doing   it  does   work;   this  is   the  external 
work.     On  the  contrary,  the  air  (or  other  substances)  may  press 
upon  the  substance,  diminish   its  bulk,  and  thus  do  external 
work  on  the  substance.     The  external  work  done  by  a  rod  in 
expanding  is  very  small,  seeing  that  its  boundary  moves  a 
very  short  distance;   the   external   work   done  by  a  gas  in 
expanding  may  be  considerable. 

(2)  The  internal  work,  we  see,  may  consist  in  an  increase 
in  the  energy  of  the  particles. 

(a)  The  atoms  of  which  the  molecule  is  composed  may 
vibrate  more    rapidly,  and  you   have  increase   of  energy  of 
vibration ;  or  they  may  change  their  rotation,  and  then  there 
is  change  of  energy  of  rotation  ;  both  being  forms  of  kinetic 
energy.     This  is  called  intramolecular  work. 

(b)  The  molecules  may  have    increased  kinetic  energy  im- 
parted to  them  by  increasing  their  average  velocity. 

(c)  The  molecules  may  have  been    separated  against  the 
force  of  cohesion,  causing  change  of  volume,  change  of  state. 
They  are  in  a   condition  such  that  they  will  tend  to  come 
together  again,  and  in  so  doing  will  do  work ;  so  that  we  have 
heat-energy  changed  into  potential  energy. 

(d)  The  heat  may  be  so  great  that  the  molecules  may  break 
up,  as  the  molecules  of  water  will  break  up  into  molecules  of 
oxygen  and  hydrogen. 

Change  in  the  temperature  of  a  body  is  due  to  (b)  and  (0), 
the  change  in  the  total  kinetic  energy  of  the  molecules.  In 


126  Heat 

the  past  experiments  it  is  that  part  of  the  heat  we  have  spoken 
of  as  sensible  heat.  This  is  only  part  of  the  internal  work. 
In  the  other  part  of  the  internal  work  (c)  the  energy  no  longer 
exists  as  heat,  (c)  may  be  partially  illustrated  by  pulling  apart 
two  balls  joined  by  indiarubber ;  work  has  been  done  upon  the 
two  balls,  and  exists  as  the  potential  energy  of  the  balls ;  by 
virtue  of  their  position  they  can  again  do  work.  It  represents 
the  so-called  latent  heat. 

Heat-energy  communicated  to  a  body  =  (a)  energy  of 
internal  work  (includes  increase  of  temperature)  -f  (b)  energy 
of  external  work  +  (c)  energy  communicated  to  external 
bodies  by  radiation  +  (d)  kinetic  energy  communicated  to 
external  bodies.  Generally  it  is  so  arranged  that  (c)  and  (d) 
are  each  so  small  that  they  can  be  neglected. 

Let  us  consider  a  few  examples,  (i)  Take  the  case  of  a 
mass  of  iron  receiving  heat-energy.  We  have — 

Internal  work.  An  increase  in  temperature,  or  increase 
in  the  kinetic  energy  of  the  molecules. 

In  expanding,  work  is  done  against  molecular  forces. 

External  work.  The  iron  expands,  and  pushes  the  sur- 
rounding air,  and  does  external  work.  This  is  so  small  it  can 
generally  be  neglected.  If  the  iron. were  in  vacuo,  so  that  it 
received  no  opposition  to  its  efforts  to  expand,  the  amount 
would  be  nil. 

(2)  Heat-energy  is  supplied  to  a  cubic  foot  of  ice  at  o°  C.  We 
have  work  done  among  the  molecules  in  fusion,  so  that  the  ice 
can  exist  as  water.  These  molecules,  in  returning  to  the  original 
condition,  will  give  up  the  same  amount  of  energy.  This  is 
the  so-called  latent  heat  of  ice,  and  we  see  that  it  is  not  heat 
at  all,  but  energy  existing  as  potential  energy.  The  cubic 
foot  of  ice  occupies  about  eleven-twelfths  cubic  foot  as  water  \ 
so  that  external  work  has  been  done  upon  that  substance  by  the 
pressure  of  the  atmosphere.  This  can  be  readily  calculated. 
Imagine  the  change  to  have  taken  place  on  one  face.  The 
pressure  of  the  atmosphere  is  (15  x  144)  Ibs.  per  square  foot 
nearly.  The  face  moves  one-twelfth  foot ;  therefore  the  work 
done  by  the  atmosphere  is — 

15  x  i44  x=  1 80  foot-pounds  of  work 


Change  of  State — The  Nature  of  Heat         127 

To  effect  this  change  the  ice  has  received  about  80  x  62-5 
thermal  units  =  5000  thermal  units,  equivalent  to  (5000  x  1390) 
foot-pounds  (Centigrade  scale)  of  work;  or  6,950,000  foot- 
pounds of  work.  In  comparison  with  this  the  180  foot-pounds 
of  external  work  done  upon  the  substance  is  negligible. 

(3)  Consider  the  case  of  a  cubic  foot  of  water  at  100°  C. 
receiving  heat  until  it  vaporizes  into  steam  at  100°  under  ordi- 
nary atmospheric  pressure.  There  is  first  the  internal  work 
done  in  changing  the  liquid  into  a  gas;  that  is,  work  done 
among  the  molecules  in  vaporization.  To  change  i  Ib.  of 
water  takes  537  thermal  units,  i  cubic  foot  weighs  about 
62*5  Ibs.  Therefore  the  energy  spent  in  the  change  is 
537  X  62-5  x  772  in  foot-pounds,  following  the  Fahrenheit 
scale;  that  is,  41,456,400  foot-pounds  of  work. 

A  cubic  foot  of  water  becomes  about  1700  cubic  feet  of 
steam.  If  we  suppose  the  water  enclosed  in  a  cylinder  with 
rigid  sides  and  base,  i  foot,  we  see  that  it  will  lift  a  piston 
placed  in  the  water  1700  feet,  and  do 
1700  X  144  X  15  foot-pounds  of  work; 
that  is,  3,072,000  foot-pounds  of  work, 
taking  the  pressure  of  the  atmosphere  as 
15  Ibs.  on  the  square  inch.  Thus  we 
have — 

Total  work    .. 
External  work 

Internal  work 


41,456,400  foot-pounds 
3,072,000 


38,384,400 

The  internal  work  is   a   little   more 
than  twelve  times  the  external  work. 

79.  Melting  Point. — Let  us  now 
return  to  fusion,  and  examine  first  the 
melting  points  of  solids.  For  substances 
whose  melting  point  is  low  it  can  be  de- 
termined by  melting  them  (wax,  paraffin) 
and  drawing  them  into  very  thin  capillary 
tubes.  Cut  off  a  piece  of  the  tubing  when  the  substance  has  set, 
and  attach  it  by  an  indiarubber  band  to  a  thermometer  (Fig.  49). 
Place  the  whole  in  a  liquid  which  boils  at  a  higher  tempera- 


FIG.  49. 


128  Heat 

ture  than  the  melting  point.     Heat  slowly,  and  observe  (i) 
the  temperature  at  which  fusion  begins,  /.    Remove  the  flame, 

and  observe  (2)  when  solidification  begins,  /.     '—^—  will  be 

the  melting  point. 

From  such  experiments  the  laws  of  fusion  have  been 
deduced. 

80.  Laws  of  Fusion.— (i)  Every  solid  substance  begins 
to  melt  at  a  certain  definite  temperature,  if  the  pressure  remains 
constant.  This  is  called  the  melting  point  (or,  in  the  reverse 
process,  the  freezing  point). 

(2)  If  the  mass  be  kept  stirred  so  as  to  ensure  uniformity 
of  temperature,  the  temperature  remains  constant  from  the 
beginning  to  the  ending  of  the  fusion. 

MELTING  POINTS  (C.°)  AT  A  PRESSURE  OF  i  ATMOSPHERE. 


Mercury 

...  -40° 

Sulphur 

..    us0 

Ice 

...           0° 

Bismuth 

...    260° 

Butter  ... 

-      33° 

Lead 

...    327° 

Phosphorus 

...      44° 

Iron  ... 

1600° 

White  wax 

...      65° 

Platinum 

...     1775° 

81.  Latent  Heat  of  Fusion. — The  number  of  units  of 
heat  required  to  change  a  unit  of  mass  of  a  solid  at  its  melting 
point  into  a  liquid  at  the  same  temperature  is  called  the  latent 
heat  of  fusion  of  the  solid. 

In  melting  the  ice  at  o°  C.  to  water  at  o°  C.  (§  70)  the  heat 
absorbed  had  no  effect  upon  the  temperature.  For  this  reason 
it  was  said  to  be  latent,  and  was  called  by  Black  the  latent 
heat  of  water.  The  heat  thus  become  latent  was  supposed  to 
be  stored  away  in  the  substance  as  heat,  and  became  evident 
as  sensible  heat  when  the  liquid  passed  into  the  solid  state. 
We  have  seen  that  the  heat  is  engaged  in  doing  internal  work, 
in  overcoming  the  cohesion  of  the  molecules  so  that  the  solid 
ice  can  exist  as  liquid  water ;  that  until  the  change  of  state 
is  completed  there  will  be  no  rise  of  temperature ;  and  that 
latent  heat  is  potential  energy. 

Water  =  ice  +  potential  energy. 


Change  of  State — The  Nature  of  Heat         129 

There  is  also  generally  a  change  in  volume.  If  the  sub- 
stance, in  liquefying,  increase  in  volume,  then  the  energy 
.as  heat  necessary  to  produce  fusion  equals  the  potential  energy 
of  the  molecules  (latent  heat)  together  with  the  external  work 
done  in  overcoming  the  pressure  of  the  atmosphere,  this  latter 
being  a  small  quantity  comparable  with  the  former  (§  78). 
If  the  volume  is  less  in  the  liquid  state,  external  work  is 
done  on  the  substance. 

82.  Black's  Methods. — Two  of  Black's  methods  possess 
an  historical  interest.  Two  similar  thin  flasks,  one  containing 
5  ozs.  of  melting  ice  at  32°F.,  the  other  5  ozs.  of  water  at 
33°  F.,  were  suspended  in  a  room  whose  temperature  (47°  F.) 
remained  constant  during  the  experiment. 

(1)  In  \  hour  the  temperature  of  the  water  rose  to  40°  F., 
the  ice-flask  still  indicating  32°. 

(2)  In  10 \  hours  the  ice  was  melted,  and  the  temperature 
of  the  flask  had  risen  to  40°  F. 

The  ice-flask  had  received  in  10^  hours  twenty-one  times 
as  much  heat  as  the  water-flask  received  in  \  hour ;  the  latter 
received  7  units  of  heat  during  that  time.  Therefore  to  melt 
5  ozs.  of  ice  at  32°  F.  to  water  at  32°  F.,  and  raise  its  temperature 
to  40°,  requires  147  units ;  that  is,  the  latent  heat  of  water  is 
139  units  (the  unit  of  mass  is  5  ozs.).  This  rough  result,  on 

the  Centigrade  scale,  will  be  139  X  -  -  77. 

His  second  method  was  a  simple  one  of  mixing  ice 
with  water,  and  from  the  resulting  temperature  determining 
the  latent  heat. 

He  took  a  piece  of  ice  weighing  59^  drachms,  and  placed 
it  in  67  J  drachms  of  water  at  190°  F.  In  a  few  seconds  the 
ice  was  melted,  and  the  final  temperature  was  53°  F.  He 
previously  determined  that  the  water  equivalent  of  the  glass 
vessel  was  4  drachms. 

.*.  67^  +  4  =  71^  drachms  of  water  fell  in  temperature  from 
190°  to  53°,  /'.<?.  through  137°,  and  gave  up  71^  X  137  = 
9795^  heat-units  (taking  i  drachm  as  the  unit  of  mass) 

(a)  59^  drachms  of  ice  at  32°  F.  were  melted  to  water  at 

K 


130  Heat 

the  same  temperature;  and  (b)  59^  drachms  of  water  were  raised 
through  21°,  i.e.  from  32°  to  53°. 

(b)  required  59^  X  21  =  1249^  heat-units 
.*.  (a)  required  9795^  —  1249^  =  8546  heat- units 

.°.  to  melt  i  drachm  of  ice  requires  8546  -4-  59^  =  144  heat- 
units 

.*.  the  latem  heat  of  fusion,  using  the  Fahrenheit  scale,  is  144 
„  „  „  Centigrade       „          80 

a  number  that  more  accurate  experiments  have  not  changed. 

To  simplify  the  calculation,  we  may  conduct  the  experi- 
ments so  that  the  final  temperature  is  the  freezing  point. 

For  example,  leaving  out  the  water  equivalent  of  the  vessel — 
i  Ib.  of  ice  or  snow  at  32°  F.,  mixed  with  i  Ib.  of  water  at 
176°  F.,  gives  2  Ibs.  of  water  at  32°. 

;.  at  once  the  latent  heat  of  fusion  =  144  (Fahrenheit  scale) 

Or  i  Ib.  of  ice  at  o°  C.,  mixed  with  i  Ib.  of  water  at  80°  C, 
gives  2  Ibs.  of  water  at  o°. 

.*.  the  latent  heat  of  fusion  is  80  (Centigrade  scale) 

83.  Laboratory  Method.— This  is  simply  Black's 
method  with  corrections.  The  calorimeter  already  described, 
provided  with  a  lid  and  a  stirrer,  can  be  used. 

Let  m  be  the  mass  of  water  at  temperature  /  (the  experi- 
ment should  be  so  arranged  that  the  temperature  at  the 
beginning  of  the  experiment  is  as  much  above  the  temperature 
of  the  room  as  it  is  below  it  at  the  end  of  the  experiment) ; 
and  CD,  the  water  equivalent  of  the  calorimeter,  thermometer, 
etc.;  M,  the  mass  of  dry  ice  added  in  pieces.  The  mass 
M  is  obtained  by  weighing  the  calorimeter  before  the  ice  is 
added  and  at  the  end  of  the  experiment.  Let  0  be  the  final 
temperature. 

The  water  has  lost  (m  +  <•>)(/  -  6)  units  of  heat.  These 
units  of  heat  have  melted  M  units  of  ice,  and  have  raised  the 
temperature  0°. 


Change  of  State — The  Nature  of  Heat        131 

If  x  be  the  latent  heat  of  water,  these  operations  require 
MX  +  M0  units  of  heat. 

.'.  MX  +  M0  =  (m  -f  <o)  (t  -  0) 

_(OT  +  q»)(/-0) 

M 

A  calorimeter  weighs  18  grams ;  after  adding  water,  the 
weight  is  168  grams,  the  temperature  (/)  being  24-5°  C. ;  some 
dry  ice  at  o°  is  added ;  after  the  whole  is  melted,  the  lowest 
temperature  reached  is  (0)  13*5°  C. ;  the  total  weight  is  now 
1 86  grams :  find  the  latent  heat  of  ice,  given  that  the  water 
equivalent  (o>)  of  calorimeter,  thermometer,  etc.,  is  3  grams. 

The  mass  of  water  {m}  is  168  —  18  =  150  grams 
„         „         ice  (M)  is  186  -  168  =  18 

(a)  The  calorimeter  and  contents  lose  (150  +  2>)(24'S  ~~  I3'S) 

=  153  X  ii  heat-units  =  1683  heat-units 

[(*  +  •)(/-*)] 

(b)  1683    heat-units    melt    18   grams   of  ice,    and   raise    the 

temperature  of  water  formed  from  o°  to  13*5°;  the  latter 
takes  18  X  13-5  =  243  heat-units  [Mx  -f  MO] 

.*.  to  melt  18  grams  requires  1683  —  243  =  144  heat-units 
.*.   to  melt  i  gram  „       1440 -f-  18  =  80  „ 

.*.  the  latent  heat  of  ice  =  80 

A  slight  modification  of  the  method  enables  us  to  calculate 
the  latent  heat  of  fusion  of  solids  that  are  solid  at  the  ordinary 
temperature.  Let  us  take,  for  example,  the  case  of  sulphur. 

The  following  particulars  will  be  required.  They  must 
either  be  taken  from  the  results  of  other  experiments  or  must 
be  determined  : — 

(1)  The  melting  point  of  sulphur,  /          =  115°  C. 

(2)  The  specific  heat  of  solid  sulphur,  c  =  0-203. 

(3)  »  „       liquid      „       <!  =  0-234. 

(4)  The  mass  of  water  in   the    calorimeter  (m)  and  the 
water  equivalent  (w).     Let  the  sum  of  these  two  (;;/  +  o>)  in  an 
experiment  =12  grams,  the  initial  temperature,  /M  being  16°  C 

The  sulphur  is  melted,  heated  to  f°  (say  150°),  and  poured 


132  Heat 

into  the  calorimeter.  After  mixing,  the  final  temperature,  0  (in 
the  experiment,  27°),  is  read.  The  increase  in  weight  of  the 
calorimeter  and  contents  gives  M,  the  mass  of  sulphur 
(4  grams). 

Let  the  latent  heat  of  fusion  for  sulphur  be  L. 

A.  The  sulphur  gives  up  heat  in  three  stages — 

(a)  Cooling   from   150°  to  115°;    heat-units    liberated   = 
4(150  -  115)  x  0-234  -  3276.  [M(/'  -  t)c\ 

(b)  Solidifying  from  liquid  at   115°  to  solid  at  115°;  heat- 
units  liberated  =  4!,.  [ML] 

(c)  Cooling  in  the  solid  state  from  115°  to  27°;  heat-units 
liberated  =  4(115  -  27)  X  0-203  =  61-32.       [M(/  -  0)c'] 

A  total  of  3276  +  4L  4-  61*32  heat-units  =  4L  +  94-08, 

[M(/(  -  fy  +  ML  -f .  M(/  -  ey\ 

B.  The  water  and  calorimeter  receive   12(27  —  J6)  heat- 
units  =  132.  [(;//  -f  <o)(0  -  fj)] 

/.  assuming  that  all  the  heat  from  the  sulphur  is  given   to 
the  calorimeter  and  water — 

4L  4-  94-08  =132 
•••  L  =  9-5 
Or  generally — 

M(/'  -  t)c  +  ML  +  M(/  -  BY  =  (m  +  <o)  (0  -  tj         (i.) 

where  L  is  the  only  unknown  term.  The  specific  heat  of  solids 
(c)  is  known  in  most  cases,  or  it  can  be  readily  determined. 
c\  the  specific  heat  of  liquids,  is  more  difficult  to  determine. 
If,  however,  we  repeat  the  above  experiment,  beginning  at 
another  temperature,  /"°,  and  ending  at  a  final  temperature  0'°, 
M'  being  the  mass  of  sulphur,  then— 

M'(/"  -  f)c  -f  M'L  +  M'(/  -  0y  =  (/;/  -f  <•>)  (9  -  /)      (ii.) 
we  have  two  equations  (i.)  and  (ii.),  and  two  unknowns,  L  and  c\ 
:.  both  of  these  can  be  determined 

Substances  heated  to  high  temperatures  lose  heat  rapidly 
by  radiation.  In  attempting  to  determine  the  latent  heat  of 
fusion  of  tin,  for  example  (melting  point  237°  C.),  the  substance 
must  be  heated  to  a  temperature  about  250°.  Corrections 


Change  of  State  —  T/ie  Nature  of  Heat        133 

must,  therefore,  be  introduced  to  allow  for  the  loss  in  trans- 
ferring it  to  the  water,  and  allowance  must  also  be  made  for 
the  loss  due  to  the  cooling  of  the  calorimeter. 

A  substance  like  paraffin,  that  is  solid  at  ordinary  tempera- 
tures, and  which  melts  below  100°  C.,  may  be  added  to  the 
calorimeter  in  the  solid  state,  the  water  being  at  a  temperature, 
/!,  above  the  melting  point,  /.  Let  the  final  temperature  be  0, 
the  temperature  of  the  solid  paraffin  /'.  Then,  using  letters 
as  in  the  above  equations  — 

(m  +  to)  ft  -  0)  =  M(/  -  t')c  +  ML  +  M(<9  -  t)c' 


Ice           

80 

Tin 

Nitrate  of  soda    .  .  . 

63 

Bismuth  ... 

Zinc          

28 

Sulphur   ... 

Platinum  

27 

Phosphorus 

Silver        

25 

Mercury  .  .  . 

LATENT  HEATS  OF  FUSION. 

...  i4'6 
...  12-6 
..  9-4 
...  5-0 
2-8 

84.  Solution.  —  Heat  has  been  used  to  fuse  solids.  They 
also  pass  into  the  liquid  state  when  placed  in  a  suitable  liquid, 
as  salt  in  water.  Heat,  again,  is  necessary  ;  this  is  taken  from 
the  heat  of  the  whole  mass,  and  the  temperature  falls. 

Place  the  two  bulbs  of  the  differential  thermometer  in 
dishes  of  water  at  the  same  temperature.  When  the  index  is 
stationary,  add  crystals  of  a  soluble  substance  (sodium  sulphate) 
to  one  dish  ;  as  it  dissolves,  the  index  will  show  that  the 
temperature  of  the  solution  is  falling.  This  is  the  principle 
of  many  freezing  mixtures.  Mixtures  of  solids,  or  of  solids 
and  liquids,  provided  that  one  dissolves,  cause  a  lowering  of 
temperature 

The  following  are  some  examples  of  such  mixtures,  the  first 
temperature  being  that  of  the  substance  before  mixing  :  — 

Ice  and  salt         ...         .........  down  to—  22°  C. 

Sodium  sulphate  (8  parts),  hydrochloric 

acid  (5  parts)          .........  from  10°  „   —  i7°C. 

Sodium  phosphate  (9  parts),  dilute  nitric 

acid  (4  parts)          .........  „     10°  „    —  29°C. 

Calcium  chloride  (4  parts),  snow  (3  parts)  ,,  p°  „  -5i°C, 


134 

In  the  above  mixtures,  the  solid  substance  used  is  solid  at 
ordinary  temperatures.  Much  lower  temperatures  have  been 
obtained  by  using  as  the  solid  a  substance  that  is  a  gas  at 
ordinary  temperatures,  but  which  has,  by  pressure  or  reduction 
of  the  temperature,  been  changed  into  the  solid  state.  This 
will  be  illustrated  when  we  speak  of  the  liquefaction  of 
gases(§n8). 

The  lowering  of  temperature  may  be  disguised  by  chemical 
action  taking  place  between  the  substances.  For  example,  if 
phosphorus  pentoxide  be  placed  in  water,  it  dissolves,  and  there 
is  a  distinct  rise  in  temperature — the  heat-units,  the  result  of 
chemical  action  between  the  substance  and  water,  being  greatly 
in  excess  of  the  heat-units  needed  to  dissolve  the  substance; 
dissolving  sodium  hydrate  or  potassic  hydrate  in  water  gives 
a  similar  result.  This  part  of  the  subject  belongs,  however, 
rather  to  chemistry  than  to  physics. 

85.  Saturation. — On  adding  carefully  a  solid  to  a  liquid 
that  will  dissolve  it,  a  point  is  reached  at  any  temperature 
when  no  more  will  dissolve :  the  liquid  is  saturated  at  that 
temperature.    Generally,  on  raising  the  temperature  more  solid 
dissolves ;   this,  however,  offers  many  exceptions.     Common 
salt,  for  example,  seems  equally  soluble  in  water  at  all  tem- 
peratures. 

86.  Solidification.— If  a  liquid  pass  into  the  solid  state, 
the  thermal  units  necessary  to  change  the  solid  into  the  liquid 
state   are   liberated.     The   potential   energy  is  changed   into 
heat.     The   freezing  point  is   the   same   temperature  as  the 
melting  point  under  the  same  pressure,  and  until  solidification 
is  completed  the  temperature  remains  constant. 

Under  certain  conditions,  the  liquid  may  be  cooled  below 
the  freezing  point  without  solidification  taking  place. 

Water,  for  example,  deprived  of  air  and  placed  in  a  clean 
capillary  tube  has  been  cooled  down  to  —20°  C.  without 
freezing.  In  ordinary  vessels,  if  they  be  not  agitated,  water 
will  remain  liquid  below  o°  C.,  but  on  shaking  or  dropping 
into  it  a  piece  of  ice,  solidification  takes  place.  In  all  cases, 
as  soon  as  solidification  begins,  the  temperature  rises  to  the 
freezing  point. 


Change  of  State — The  Nature  of  Heat         135 

The  phenomenon  is  readily  illustrated  by  making  a  strong 
solution  of  sodium  sulphate,  evaporating  it  until  a  few  drops 
on  cooling  crystallize.  If  the  solution  be  poured  into  a  clean 
glass  flask,  a  thermometer  inserted,  a  piece  of  cotton  wool  be 
placed  in  the  neck,  and  the  flask  be  placed  so  that  it  is  not 
disturbed,  the  solution  will  cool  down  to  the  temperature  of 
the  room  without  solidifying.  If  a  crystal  of  sodium  sulphate 
be  dropped  into  it,  the  mass  crystallizes,  and  the  temperature, 
as  shown  by  the  thermometer,  rises. 

Sodium  hyposulphate,  whose  crystals  melt  readily  in  their 
own  water  of  crystallization  on  placing  the  vessel  containing 
them  in  hot  water,  is  also  suitable  for  the  experiment 

If  the  substance  be  dissolved  in  a  suitable  solvent,  the  solid 
is  formed  on  cooling  the  mixture  sufficiently.  The  effect  of 
the  presence  of  the  substance  is  to  reduce  the  freezing  point  of 
the  liquid.  For  example,  if  salt  be  dissolved  in  water,  the  mixture 
remains  liquid  at  temperatures  below  o°  C.  A  familiar  illustra- 
tion occurs  in  the  liquid  formed  by  adding  salt  to  the  snow  on  the 
tram-lines  in  winter  :  the  snow  dissolves,  but  a  liquid  mixture  of 
water  and  salt  is  formed,  much  colder  than  the  snow. 

Blagden,  in  1788,  stated  that  the  lowering  of  the  freezing 
point  was  proportional  to  the  amount  of  salt  dissolved.  This 
is  known  as  Blagden 's  law,  and  is  practically  true  for  weak 
solutions. 

Raoult  has  shown,  further,  that  this  law  is  applicable  to 
most  solutions,  provided  the  mass  of  salt  be  not  more  than 

-  of  the  mass  of  the  liquid.      The   researches   of  Raoult 

100 

suggest  that  some  simple  general  law  may  be  deduced  for 
solutions. 

87.  Guthrie's  Researches. — Cryohydrates.— If  we 
reduce  the  temperature  of  a  weak  mixture  of  water  and  salt  (to 
take  a  special  example),  then,  at  a  temperature  below  zero 
depending  upon  the  proportion  of  salt,  solidification  of  part  of 
the  pure  water  takes  place,  and  therefore  a  stronger  solution 
of  brine  is  left.  This  fact  was  known  to  the  earlier  experimenters. 

Guthrie,  experimenting  upon  strong  solutions,  has  shown  that 
at  first  crystals  are  deposited  that  have  a  definite  composition, 


136 


Heat 


and  contain  two  molecules  of  water  to  one  of  salt  (NaCl,2H2O) ; 
if  these  crystals  be  removed,  and  the  temperature  be  further 
lowered,  the  whole  mass  freezes  at  about  -  2 1°  C.,  and  the  solid 
has  a  definite  composition  of  twenty-one  molecules  of  water  to 
two  of  salt  (2NaCl,2iH2O).  To  these  compounds  he  gave  the 
name  cryohydrates. 

The  cryohydrates  are  definite  compounds  that  possess  a 
definite  melting  point  and  crystallize  in  definite  forms. 

The  cryohydrate  represents  the  mixture  that  will  the 
longest  resist  solidification.  In  the  case  of  water  and  salt,  the 
lowest  possible  temperature  to  which  a  given  mixture  can  keep 
solid  is  —22°  C.,  and  this  is  possible  when  a  mixture  is  made 
represented  by  2NaCl,2iH2O. 

88.  Change  of  Volume.— Change  of  state  is  in  general 
accompanied  by  change  of  volume.  The  greater  number  of 
solids  expand  when  liquefied,  and  therefore  the  solid  sinks 
in  the  liquid ;  examples  are  gold,  lead,  paraffin,  wax.  Cast 
iron,  ice,  bismuth,  antimony,  contract  on  liquefying ;  the  solid, 
therefore,  floats  on  the  liquid.  Castings  taken  from  the  former 
lack  the  sharpness  of  those  taken  from  the  latter  set  of  solids. 
Gold  coins  are  therefore  stamped. 


Substance. 

Melting  point. 
C. 

Density  of 
solid. 

Density  of 
liquid. 

Bismuth            

260° 

9-82 

10-05 

Lead      

327° 

II'4 

I0'4 

Iron       

I600° 

6'95 

6-88 

Water    

0° 

0*9167 

I'O 

The  case  of  water  is  important  in  nature.  1 1  cubic  feet  of 
water  at  o°  become  roughly  1 2  cubic  feet  of  ice  at  o°.  Therefore 
the  ice  formed  on  ponds  floats  on  the  surface  and  retards  further 
freezing  ;  otherwise  the  lakes  and  ponds  and  rivers  would 
become  a  mass  of  solid  ice,  that  in  many  cases  the  sun  of 
summer  would  be  unable  to  melt. 

(a)  In  the  case  of  ice,  cast  iron,  etc.,  we  have  heat-energy 
=  internal  work  —  external  work  done  on  the  solid  in  con- 
tracting. 


Change  of  State — The  Nature  of  Heat         137 

(b)  In  the  case  of  gold,  paraffin,  etc.,  heat-energy  =  in- 
ternal work  -f-  external  work  done  in  expanding. 

This  would  suggest  that  increase  or  decrease  of  pressure  in 
these  two  classes  would  lead  to  different  results. 

In  class  (a)  increase  of  pressure  lowers  the  melting  point. 
In  class  (b)  increase  of  pressure  raises  the  melting  point  (see 
§  91  and  chapter  xii.). 

It  was  predicted  by  Professor  T.  Thomson,  and  confirmed 
by  experiment  by  Sir  W.  Thomson,  that  increase  of  pressure 
lowers  the  melting  point  of  ice.  The  melting  point  is  lowered 
°'0075°  C.  per  atmosphere.  In  the  case  of  paraffin,  on  the 
other  hand,  which  ordinarily  melts  at  46-3°  C,  it  melts  at 
49*9°  C.  when  subject  to  a  pressure  of  100  atmospheres. 

89.  Ice. — The  work  done  by  ice  in  expanding  against  the 
atmosphere  is  not  observable,  but  if  it  be  enclosed  in  strong 
cases,  as  in  water-pipes,  we  have  burst  pipes  due  to  expansion, 
the  flaws  being  made  evident  during  the  thaw. 

The  force  due  to  this  expansion  is  strikingly  shown  in 
the  experiments  of  Major  Williams  in  Canada.  Bombs  were 
filled  with  water,  closed,  and  exposed  to  the 
frost.  On  freezing,  the  plugs  were  forced  out 
and  the  bombs  split ;  a  cylinder  of  ice  issued 
from  the  opening  wherever  made  (Fig.  50). 
:  The  lowering  of  the  temperature  would 
cause  the  water  to  expand  in  attempting 
to  pass  into  ice,  this  would  increase  the  pressure,  and  probably 
lower  the  melting  point,  so  that  in  bursting  a  stream  of  water 
issued.  The  pressure  being  relieved,  and  the  temperature 
being  below  the  temperature  at  which  water  freezes  under 
ordinary  pressure,  the  water  at  once  froze. 

90.  Regelation. — Faraday  drew  attention  to  the  fact  that, 
when   two  pieces  of  ice  are  pressed   together,  they   freeze 
together  at  the   place  of  contact.      Bottomley  suspended   a 
block  of  ice  upon  two  supports,  passed  a  wire  round  the  block, 
and  attached  a  weight  to  it;  the  wire  gradually  cut  its  way 
through,  the  block  freezing  as  the  wire  passed.     The  explana- 
tion is  that  the  increased  pressure  lowers  the  melting  point, 
and  part  melts.     The  water  occupies  less  volume,  the  pressure 


138  Heat 

is  thus  relieved,  the  water  readily  escapes4  from  the  place,  and 
the  pressure  on  it  being  relieved,  and  it  being  below  freezing 
point,  it  again  freezes. 

This  is  seen  on  a  large  scale  in  the  motion  of  glaciers. 
In  the  slow  yet  river-like  motion  there  will  be  enormous 
pressure  down  the  mass.  At  particular  points — the  convex 
side  of  a  bend,  for  example — the  pressure  will  be  enormously 
increased.  The  general  temperature,  save  at  the  surface,  will 
not  be  much  below  o°,  as  there  is  a  continuous  trickle  of 
water.  The  melting  point  will  be  lowered,  and  part  of  the 
ice  will  melt.  The  water  takes  up  less  volume,  and  there  is 
thus  a  relief  at  once ;  the  water  flows  away  and  is  at  once 
frozen.  Thus  the  mass  acts  as  if  it  had  the  power  of  bending, 
or  as  if  its  nature  were  viscous  or  plastic. 

91.  Melting  Point  of  Ice  lowered  by  Pressure. — That 
the  melting  point  is  lowered  by  pressure  is  well  illustrated  by  a 
piece  of  apparatus  first  made  by  Mousson.  A  strong  cylinder 
of  steel  is  closed  at  one  end  with  a  screw  and  at  the  other 
with  a  screw-piston  provided  with  a  handle.  On  turning  the 
handle  the  piston  moves  in  the  tube.  The  piston  is  inserted, 
the  apparatus  inverted  and  nearly  filled  with  water,  a  small 
piece  of  copper  placed  in  the-  water  sinks  until  it  is  in 
contact  with  the  piston.  The  whole  is  plunged  into  a  freezing 
mixture  until  the  water  freezes.  The  screw  is  then  placed  on 
the  open  end,  and  the  whole  again  inverted,  so  that  we  have  a 
cylinder  of  ice  with  the  piece  of  copper  on  the  top  of  it,  and  the 
screw-piston  above  the  piece  of  copper.  The  whole  apparatus 
is  kept  at  a  temperature  of  - 18°  C.,  and  the  handle  turned  so 
that  the  piston  presses  upon  the  ice.  When  the  pressure  exerted 
was  equal  to  13,000  atmospheres,  the  pressure  was  relieved. 
On  taking  out  both  screw-piston  and  screw-stopper,  it  was 
found  that  the  piece  of  copper  was  resting  upon  the  screw- 
stopper. 

The  temperature  was  - 18°  C.  throughout.  Evidently  at  a 
pressure  near  13,000  atmospheres  the  ice  had  melted,  so  as  to 
allow  the  copper  to  sink  from  one  end  to  the  other ;  that  is, 
at  a  pressure  about  13,000  atmospheres  the  melting  point  was 
lowered  18  degrees  Centigrade. 


Change  of  State — The  Nature  of  Heat        139 

A  laboratory  experiment,  due  to  Tyndall,  can  be  performed 
with  two  box-wood  moulds.  A  convenient  form  is  that  in 
which  one  contains  a  hemispherical  cavity,  while  the  other  has 
a  hemispherical  protuberance  that  does  not  quite  fill  the  cavity. 
On  placing  broken  pieces  of  ice  or  snow  in  the  cavity,  placing 
the  second  block  above,  and  compressing  either  by  a  Bramah 
press  or  by  suitable  screws,  the  whole  can  be  moulded  into  a 
cup-shaped  piece  of  ice. 

Pieces  of  ice  or  snow  can  be  made  into  a  solid  block  of 
ice  by  placing  them  in  a  steel  mortar,  inserting  a  close-fitting 
pestle,  and  driving  the  pestle  down  with  a  hammer. 

The  whole  question  is  further  illustrated  in  making  a  snow- 
ball. If  the  temperature  of  the  air  be  below  o°  C,  the  snow 
will  not  join,  and  the  boy  cannot  exert  sufficient  pressure  to 
lower  its  melting  point  so  as  to  produce  regelation.  The 
warmth  of  the  hand  raises  its  temperature  nearly  to  melting 
point,  and  then  the  available  pressure  is  sufficient 

92.  Fusion  of  Alloys. — The  melting  point  of  an  alloy 
is  generally  lower  than  that  of  the  metals  forming  the  alloy. 
This  is  especially  characteristic  of  alloys  of  lead  and  bismuth, 
as  will  be  seen  from  the  following  table  : — 

Parts  by  weight. 

Lead  (326°  C.)  ...     i       ...  .  i  ...       i  ...  2 

Bismuth  (267°  C.)    o       ...       o  ...       2  ...  5 

Tin  (238°  C.)   ...     5       »        3  -       i  -  3 

Melting  points      i94°C...  187°  ...  9401  ...  100° 

The  low  melting  points  of  these  alloys,  combined  with  the 
fact  that  they  expand  on  solidifying,  make  them  suitable  for 
casts  and  stereotyping ;  they  are  also  used  for  solder. 

When  an  alloy  is  gradually  heated,  it  generally  softens  and 
becomes  pasty  some  degrees  below  the  point  at  which  it 
liquefies.  For  example,  an  alloy  of  two  parts  of  tin  and 
two  parts  of  lead  begins  to  soften  at  185°  C,  and  finally 
liquefies  at  189°  C.  The  reverse  process  is  observable  as  an 
alloy  cools.  There  is  first  the  passage  from  a  liquid  into  the 
pasty  condition ;  this  takes  some  time,  the  temperature  re- 

1  Rose's  fusible  metal,  which  melts,  therefore,  in  hot  water. 


I4O  Heat 

maining  stationary,  and  latent  heat  being  liberated.  Then 
follows  the  cooling  through  this  viscous  state,  and  the  final 
solidification  at  the  lower  temperature. 

Rudberg  studied  various  alloys  of  lead  and  tin,  and  noted 
how  long  it  took  them  in  a  liquid  state  to  cool  through  10°  C. 
Whatever  proportions  of  the  metals  were  used,  he  noticed 
that  the  rate  of  cooling  was  slowest  near  187°  C.,  the  temperature 
at  which  they  finally  solidified.  This  was  the  temperature 
at  which  the  alloy  3  of  tin  to  2  of  lead  solidified  without 
passing  through  the  viscous  state.  He  noticed  a  similar 
phenomenon  in  other  mixtures,  and  concluded  that  an  alloy 
might  be  considered  as  a  mixture  of  two  definite  alloys,  each 
with  its  own  melting  point;  that  the  alloy  with  the  lower 
melting  point  first  began  to  melt ;  the  tendency  of  the  mixture 
to  flow  would  depend  upon  the  proportion  of  this  alloy. 
Finally,  when  the  higher  melting  point  was  reached,  the 
whole  liquefied.  For  mixtures  of  lead  and  tin,  the  lower 
melting  point  was  always  187°,  the  melting  point  of  the  alloy 
i  of  lead  to  3  of  tin  ;  this  he  called  a  chemical  alloy.  This 
definite  lower  melting  point  does  not,  however,  seem  to  be  as 
definite  as  Rudberg's  conclusions  suggest. 

Mixtures  of  lead  and  tin  are  used  as  fusible  plugs  in  boilers. 
Steam  at  each  pressure  has  a  definite  temperature  (§  97). 
Suppose  the  safe  pressure  in  a  boiler  must  not  exceed  180  Ibs. 
on  the  square  inch,  the  temperature  of  the  steam  will  be  373°  F. 
If  a  fusible  plug  made  of  2  tin  and  2  lead  be  inserted,  at 
365°  F.  the  plug  will  soften,  and  at  372°  F.  will  melt,  and 
then  the  steam  will  blow  off. 

The  proportions  of  the  alloy  for  each  temperature  are 
determined  by  experiment. 

Fusible  plug.  Softens  at  Melts  at 

2  tin  2  lead  365°  F 372°  F. 

2   „  6     „  372°  F.     383°  F. 

2   „  7     „  377*°  V 388°  F. 

2   „  8     „  395°  F 408°  F. 

According  to  Rudberg,  the  number  in  column  2  should 
be  368  in  all  cases,. 


Change  of  State — The  Nature  of  Heat        141 

93.  Viscous  Solids. — Iron  has  not  any  definite  melting 
point ;  it  gradually  softens,  and  in  this  condition  can  be 
welded,  and  ultimately  melts.  Glass  passes  through  a  similar 
viscous  state.  It  may  be  that  this  is  a  property  of  many  solids, 
although  the  range  of  temperature  for  which  they  remain 
viscous  may  be  so  small  that  it  escapes  detection. 

WORKED  EXAMPLES. 

1.  A  pound  of  ice  at  0°  C.  is  thrown  into  61bs.  of  water  at  15°  C.  con- 
tained in  a  copper  vessel  weighing  3lbs»,  and  when  the  ice  is  melted,  the 
temperature  of  the  water  is  2°  C.  :  find  the  latent  heat  of  fusion  of  ice,  the 
specific  heat  of  copper  being  0*095.     (London  Matric.,  1880.) 

The  water  equivalent  of  the  vessel  is  0-095  X  3  =  0*285  Ik. 

/.  water  +  calorimeter  are  equivalent  to  6-285  Ibs. 
.'.  they  give  up  to  the  ice  (6-285X15  —  2)  =  81705  thermal  units 
2  of  these  are  used  in  raising  the  melted  ice  at  o°  to  2°  C. 
.*.  81705  —  2  =  79705  heat-units  must  be  used  in  melting  I  Ib.  of  ice  from 
ice  at  o°  to  water  at  o°  C. 

/.  latent  heat  of  fusion  =  797 

2.  20  grams  of  ice,  contained  in  a  copper  vessel  weighing  20  grams,  is 
placed  in  a  freezing  mixture  whose  temperature  is  -  20° ;  vessel  and  ice 
are  now  plunged   into   water   contained   in  a  calorimeter  whose  water 
equivalent  is  5  grams,  the  temperature  of  the  calorimeter  being  60°  C.  ; 
the  final  temperature  is  required  to  be  10°  C.  :  find  the  mass  of  the  water. 
The  mean  specific  heat  of  ice  =  0*5. 

(a)  Heat  required  to  raise  tempera- j  _  |2o(2O  X  0-5  +  20  X  0-095) 

ture  of  ice  and  copper  to  o°        )  ~  {     =20(10+ 1-9)  =  238  heat-units 

(^)  Heat  required  to  melt  the  ice  to"l 

water  at  Oo  }  =  20  X  80  =  1600  heat-units 

(c)  Heat  required  to  raise  the  water \  _  J2O  X  10  +  20  X  10  X  0^095 
at  o°  and  copper  at  o°  to  10°     /  "~  \     =  200  +  19  =  219  heat-units 

.*.  total  =  238  +  1600  -f  219  =  2057  heat-units 
(rf)  Calorimeter,  in  cooling  from  60°  to  10°,  supplies  5  X  50  =250  heat-units 

.*.  2057  —  250  =  1807  heat-units  must  come  from  the  water 
Each  gram  will  supply  50  heat-units. 

.*.  mass  of  water  = =  36*12  grams 

3.  The  density  of  ordinary  phosphorus  is  1*83  ;  it  melts  at  44°  C.,  and 
unit  volume  becomes  I  "043  ;  its  latent  heat  of  fusion  is  5  :  compare  the 


142  Heat 

external  work  with  the  internal  work  during  liquefaction.     The  pressure 
of  the  atmosphere  =  14*7  Ibs.  per  square  inch. 

Imagine  I  cubic  foot  of  solid  phosphorus  just  covered  with  water  to 
prevent  the  oxidizing  action  of  the  air.  Let  the  expansion  take  place  on 
one  face  only. 

(1)  5  units  of  heat  are  equivalent  to  5  X  1390  =  6950  foot-pounds  of 
work  =  total  work. 

(2)  The  atmosphere  is  pushed  back  through  0*034  foot. 

.*.  external  work  =  144  X  147  X  0-034  =  71*97  foot-pounds 
.*.  internal  work  =  6950  —  71*97  =  6878*03 

internal  work        7^97         l  •         i 

.  .  -  -p  =  -,!0  0       =  —  approximately 

external  work      6878  03      95    * 

4.  A  leaden  bullet  weighing  2  ozs.  strikes  a  target  with  a  velocity  of 
looo  feet  per  second  ;  its  temperature  is  16°  C.  :  if  two-thirds  of  the  energy 
of  the  bullet  be  used  in  raising  its  temperature,  determine  the  final  tempera- 
ture. (Specific  heat  of  lead  =  0*032  ;  melting  point,  326°  C.  ;  g  =  32.) 


The  kinetic  energy  =  I  *  g*^^30*   =  ^  foot-poundals 

i  thermal  unit  =  1390  foot-pounds  =  13900-  foot-poundals 

i  o8 
/.  total  kinetic  energy  is  equivalent  to  ^  x  1390  x  32  =  I>41  thermal  unit 

.*.  f  of  1*41  =  0'94  thermal  unit  is  used  in  heating  the  bullet 


To  raise  the  temperature  of  bullet  i°  C.  requires  0*032  x  —  =  0*004  th.  u. 

/.  final  temperature  =  16  -f-  0-94  -*•  0*004 
=  16  +  235  =  25i°C. 

5.  We  can  readily  determine  the  velocity  a  bullet  must  have  in  order 
to  fuse  it.     Let  the  mass  be  I  gram.     (Specific  heat  =  0-031.) 

(1)  To  raise  its  temperature  from  16°  to  326°  C.  requires  310  X  0*031 
=  9-61  thermal  units. 

(2)  To  fuse  it  requires  5*4  (latent  heat  of  fusion  of  lead)  thermal  units. 

9-61  +  5*4  =  15  thermal  units  are  equivalent  to  15  X  42,400  X  g  ergs 
=  15  X  4-2  X  io7  ergs 

If  v  be  the  velocity  of  the  body  at  impact-* 

The  kinetic  energy  =  ^mv*  =  —  -  —  ergs 

/.  v*  =  30  x  4-2  x  io7  =  12-6  x  io8 
.".  v  =  \/I2'6  x  io4cm.  per  second 

=  3-56  X  io4  =  35,600  cm.  per  second 

=  356  metres  per  second 


Change  of  State — The  Nature  of  Heat         143 

EXAMPLES.    VI. 

i.  Can  ice  at  32°  ¥.,  and  at  the  ordinary  atmospheric  pressure,  have 
its  temperature  raised  still  higher  ?     If  not,  why  not? 

2.  The  mechanical  equivalent  of  heat  is  1390  foot-pounds  Centigrade. 
What  does  this  statement  mean  ?    If  the  standard  substance  were  iron 
(whose  specific  heat  is  0*114)  instead  of  water,  what  would  be  the  value 
of  the  mechanical  equivalent  of  heat  ? 

3.  Find    the    resulting    temperature    (Centigrade)    in    the   following 
mixtures: — 

(a)  5  Ibs.  of  snow  at  o°  with  20  Ibs.  of  water  at  30°. 

(b)  3  Ibs.  of  ice  at  — 10°  with  30  Ibs.  of  water  at  60°. 

(c)  8  Ibs.  of  iron  at  200°  with  2  Ibs.  of  ice  at  o°. 

(d)  i  gram  of  phosphorus  at  o°  with  13  grams  of  water  at  25°. 

4.  How  many  heat-units  are  needed  to  change  ? — 

(a)  3  Ibs.  of  ice  at  ~5°C.  into  water  at  10°, 

(b)  4  Ibs.  of  sulphur  at  50°  C.  into  liquid  sulphur  at  300°. 
Express  the  results  in  units  of  energy. 

5.  The  latent  heat  of  fusion  of  ice  is  79 '5.      The  specific  gravity  is 
0-917.     10  grams  of  metal  at  100°  C.  are  immersed  in  a  mixture  of  ice 
and  water,  and  the  volume  of  the  mixture  is  found  to  be  reduced  to  125 
c.mm.  without  change  of  temperature:  find  the  specific  heat  of  the  metal. 

6.  The  specific  heat  of  iron  is  o'H3:    how  many  pounds  of  iron  at 
250°  C.  must  be  introduced  into  an  ice  calorimeter  to  produce  2  Ibs.  of 
water  ? 

7.  What  do  you  mean  by  external  and  internal  work  as  applied  to  a 
substance?     i  Ib.  of  copper  at  o°  C.  is  raised  to  30°  C.,  (a)  in  a  vacuum  ; 
(I))  under  ordinary  atmospheric  conditions ;  (c)  when  under  a  pressure  of 
loo  Ibs.  on  the  square  inch.     Under  which  condition  will  it  require  most 
heat  ?     Give  reasons  for  your  answer. 

8.  Ice  melts  at  32°  F.,  and  wax  at  140°  F.     A  mass  of  ice  at  31°,  and 
a  mass  of  wax  at  139°  F.,  are  separately  compressed  by  suitable  means. 
Could  either  of  these,  by  a  sufficient  increase  of  pressure,  be  melted  ?    Give 
reasons  for  your  answer. 

9.  Explain  the  phenomena  of  regelation,  and  illustrate  your  explanation 
from  glaciers  and  the  ice  formed  by  the  wheels  of  vehicles  passing  over 
snow. 

10.  Find  the  number  of  heat- units  required  to  change  i  Ib.  of  ice  at 
— 10°  to  steam  at  160°  C.     How  would  you  arrange  your  apparatus  to 
have  steam  at  such  a  temperature  ? 

11.  50  Ibs.  of  iron  fall  freely  through  a  height  of  150  feet :  if  the  kinetic 
energy  be  entirely  converted  into  heat,  and  the  heat  be  used  in  raising  the 
temperature  of  the  iron,  find  the  rise  in  temperature. 


144 


CHAPTER  VII. 

CHANGE   OF  STATE— VAPORIZATION  AND   CONDEN- 
SA  TION. 

94.  Vaporization. — The  change  of  a  liquid  into  its 
vapour,  and  the  fixity  of  the  boiling  point,  have  been  referred 
to  in  §  70,  where  water  was  treated  as  a  typical  liquid.  The 
process  of  changing  a  substance  from  the  liquid  state  into 
the  gaseous  state  is  called  vaporization,  the  reverse  process 
being  called  condensation.  Part  of  the  liquid  vaporizes  before 
the  boiling  point  is  reached;  the  gradual  disappearance  of 
water  in  vessels  exposed  to  the  air,  and  the  drying  up  of  rain  in 
the  streets,  are  everyday  phenomena.  The  change,  however, 
goes  on  much  more  rapidly  at  the  boiling  point. 

In  liquids  the  molecules  are  not  restricted  in  their  path  as 
is  the  case  with  solids.  They  impinge  one  upon  the  other, 
but  the  force  of  cohesion  has  been  so  far  overcome  that  they 
slip  readily  over  each  other,  and  there  is  no  average  position 
that  they  occupy.  The  average  velocity  of  the  molecules 
of  the  liquid  will  be  less  than  the  average  when  changed 
into  vapour,  but  some  may  have  a  velocity  greater  than 
that  of  the  vapour;  when  such  molecules  come  to  the  free 
surface  separating  the  liquid  and  its  vapour,  they  may  break 
through  the  surface  and  mingle  with  the  molecules  of  vapour. 
This  is  vaporization.  The  molecules  at  the  same  time  are 
further  separated  and  their  potential  energy  is  increased. 
Molecules  of  the  vapour  may  in  a  similar  way  rush  into  the 
liquid,  when  we  have  condensation.  Both  actions  take  place 
probably  at  the  same  time,  the  excess  of  either  determining 
under  which  phenomenon  the  process  shall  be  classed;  the 


Change  of  State —  Vaporisation  and  Condensation    14$ 

maximum  excess  of  molecules  evaporating  over  those  con- 
densing takes  place  at  the  boiling  point. 

To  effect  the  change  in  the  vibrations  and  the  speeds  of 
the  molecules,  energy  is  needed;  the  amount  is  measured 
by  the  internal  work.  All  substances  increase  in  volume 
when  they  change  into  vapour.  If  in  such  increase  they  do 
work,  as,  for  example,  in  overcoming  the  pressure  of  the 
atmosphere,  then  energy  is  also  needed  to  do  external  work. 

When  vaporization  takes  place  at  the  free  surface  only,  it 
is  termed  "  evaporation."  This  takes  place  at  all  temperatures, 
and  it  is  to  evaporation  that  the  change  of  the  greatest  amount 
of  water  into  vapour  is  due  in  nature. 

95.  Vaporization  in  Vaeuo. — The  general  study  is  best 
begun  by  an  examination  of  the  phenomenon        ABC 
of  vaporization  in  vacuo. 

Three  tubes  about  32  inches  long  are  filled 
with  mercury  and  inverted  in  the  ordinary  way, 
so  that  a  Torricellian  vacuum  is  formed  in  each 
(Fig.  51).  One  (A)  remains  during  the  experi- 
ment, and  serves  as  a  barometer.  Small  quanti- 
ties of  volatile  liquids  are  forced  up  the  other 
tubes  by  the  aid  of  a  pipette.  Take  the  case 
of  water  (B).  The  small  drop,  when  it  reaches 
the  vacuum,  at  once  evaporates,  and  the  level 
of  B  is  depressed.  The  vapour  exerts  a  pres- 
sure, measured  in  inches  or  millimetres  oi 
mercury,  by  the  difference  of  the  levels  of  A 
and  B.  As  more  water  is  allowed  to  rise,  a 
further  depression  takes  place,  until  under  the  FlG>  5I- 
conditions  of  temperature  no  more  will  evaporate,  and  a  layer 
of  water  forms  on  the  top  of  the  mercury.  The  addition  of 
further  water  does  not  perceptibly  increase  the  depression 
(neglecting  the  pressure  of  the  mass  of  water  itself) ;  there  is 
thus  for  the  particular  temperature  a  maximum  pressure  of 
water  vapour. 

If,  for  example,  the  temperature  of  the  room  be  10°  C., 
the  mercury  in  B  will  be  9  mm.  below  the  level  of  A,  that 
is,  the  maximum  pressure  of  water  vapour  at  10°  is  equal  to 


146 


Heat 


the  pressure  of  a  column  of  mercury  9  mm.  high.     If  the  tem- 
perature be  20°,  the  maximum  pressure  is  17  mm. 

Time  must  be  allowed  for  the  tube  B  to  return  to  the 
temperature  of  the  air  in  each  experiment,  because  the 
heat-energy  necessary  to  produce  evaporation  will  be  taken 
from  the  water  and  the  vapour,  and  the  temperature  will 
fall. 

If  the  tube  be  depressed  in  a  deep  trough  (Fig.  52),  as 
long  as  there  is  a  space  containing 
vapour  above  the  mercury,  the  height 
above  the  mercury  column  remains  con- 
stant (this  must  be  done  slowly,  other- 
wise heat  is  generated) ;  more  water 
vapour  condenses.  If  the  tube  be  raised 
so  that  the  top  of  B  is  far  above  A  (Fig. 
51),  the  level  still  remains  constant  as 
long  as  there  is  a  layer  of  water ;  more 
water  simply  evaporates. 

96.  Maximum  Pressure  of 
Vapours. — The  vapour  in  this  con- 
dition— that  is,  when  in  contact  with 
its  liquid— is  called  "saturated  vapour." 
The  vaporization  just  equals  the  con- 
densation in  any  one  of  the  above  posi- 
tions. If  the  space  above  the  liquid  be 
increased,  more  liquid  vaporizes,  until 
there  is  again  equilibrium  between  vapo- 
rization and  condensation. 

By  surrounding  the  tube  B  (Fig.  51) 
with  a  jacket,  through  which  passes 
water  at  various  temperatures,  and 
noting  the  difference  between  the 
heights  of  the  mercury  in  A  and  B,  the 
maximum  pressure  of  water  vapour 
at  various  temperatures  between  o°  and  100°  C.  has  been 
calculated. 

By  bending  the  tube,  and  dipping  the  end  in  freezing  mix- 
tures at  various  temperatures  (Fig.  53),  it  has  'been  shown  that 


FIG..  52- 


Change  of  State —  Vaporization  and  Condensation    147 

even  below  zero  water  vapour  exerts  a  pressure,  its  maximum 
pressure  being  definite. 

The  pressure  in  C  is  that  due  to  the  temperature  of  B, 
indicated  by  the  thermometer  /;  because  the  vapour  from  a 
layer  of  liquid  in  C  is  condensed  in  B, 
and  the  pressure  in  B  is  reduced,  more 
vapour  evaporates  from  C,  to  be  further 
condensed  in  B.  This  continues  until 
a  state  of  equilibrium  is  reached, 
when  the  pressure  in  C  is  equal  to  that 
in  B. 

97.  Regnault's  Method.— Tak- 
ing advantage  of  the  fact,  to  be  shown 
later,  that  when  a  liquid  boils  the 
pressure  of  the  vapour  is  equal  to  the 
pressure  to  which  it  is  subjected, 
Regnault  has  calculated  the  maximum 
pressure  of  aqueous  vapour  from  50° 
to  100°  C.  and  upwards. 

His  method  was  to  boil  water  in  a 
vessel,  and  to  subject  it  to  various 
pressures.  It  was  then  only  necessary 
to  note  the  temperature  when  boiling 
took  place.  For  example,  when  the 
pressure  in  the  vessel  was  39*27  Ibs. 
on  the  square  inch,  the  temperature 
of  the  water  and  steam  in  the  vessel 
when  boiling  took  place  was  266°  F. 
Therefore  the  pressure  of  water  vapour  at  a  temperature  of 
266°  F.  is  39-27  Ibs.  on  the  square  inch. 

One  form  of  the  experiment  is  shown  in  Fig.  54. 

Pure  water  was  boiled  in  a  copper  vessel,  C,  fitted  with 
delicate  thermometers.  The  steam  passed  along  the  tube 
A  B,  and  was  condensed  by  the  condenser  D,  supplied  with 
cold  water  from  E.  The  condensed  water  flowed  back  to  C, 
and  thus  the  operation  could  be  continued  without  renewing 
the  water.  A  B  was  connected  with  a  glass  globe,  M,  con- 
taining air  kept  at  a  temperature  nearly  that  of  the  outside  air 


FIG.  53. 


148 


Heat 


by  water  in  the  vessel  K.  M  was  connected  with  a  mano- 
meter, O;  M  could  also  be  connected  by  the  leaden  tube 
H  H  with  either  an  air-pump  or  a  force-pump.  -\ 

(i)  Temperature  below  1 00°  C. — H  H  was  connected  with  an 
air-pump,  and  exhausted.    The  pressure  was  indicated  by  an 


FIG.  54- 

open  manometer ;  it  would  be  the  pressure  of  the  atmosphere 
less  the  difference  in  the  lengths  of  the  two  columns  of  mercury. 
C  is  gradually  heated,  and  boiling,  known  by  the  sound  and 
the  steadiness  of  the  thermometer,  takes  place  below  100°.  A 
small  quantity  of  air  is  admitted,  the  pressure  changes,  and 
another  determination  is  made,  and  so  on  up  to  i  atmosphere. 
(2)  Above  100°. — Air  is  forced  into  M.  The  mercury  rises 
in  the  left-hand  tube,  as  in  the  figure  ;  the  pressure  =  pressure 
of  atmosphere  +  pressure  due  to  difference  in  the  columns  of 
mercury.  The  boiling  point  is  again  observed.  Such  an 
arrangement  would  only  serve  for  pressures  to  about  i£  atmo- 


60°.  148-70  „ 

112*2°.   l£ 

80°.  354*64  ,, 

120-6°.   2 

98°.  707-26  „ 

i33'9°-  3 

99°.  733'9i  » 

152*2°.  5 

1  00°.  760*00  „ 

180-3°.  10 

Change  of  State —  Vaporization  and  Condensation    149 

sphere ;  for  higher  pressures  stronger  apparatus  was  constructed, 
but  the  principle  of  the  method  remained  the  same. 


MAXIMUM  PRESSURE  OF  AQUEOUS  VAPOUR  IN  MILLIMETRES 
OF  MERCURY. 

— 10°  C.   2 -08  mm.     50°.     9 1 -98  mm.     100°.        i  atmosphere 

-4°.       3'39 

-2°.       3'95 

o°.       4-60 

10°.       9*20 

20°.     17-40 

In  the  following  table  the  temperatures  are  given  in  degrees 
Fahrenheit,  and  the  pressure  in  pounds  per  square  inch  :— 

110°.     1-267  200°.     11*52  240°        25-00 

130°.       2*212  212°.        1470  300°  67-22 

150°        3707  220°.        T7-20  350°.        I35TI 

Complete  tables,  from  which  these  are  extracted,  are  of  the 
utmost  importance  in  questions  relating  to  the  strength  of 
steam  boilers. 

By  experimenting  similarly  with  alcohol,  ether,  and  other 
volatile  liquids,  their  maximum  pressures  at  various  temperatures 
can  be  determined  (use  C,  Fig.  51).  Their  pressures  are  much 
greater  than  water.  Thus  if  ether  were  used,  and  the  tubes 
were  at  o°,  the  mercury  would  be  depressed  182  mm. ;  alcohol 
at  the  same  temperature  would  give  a  depression  of  13  mm. 
At  10°  C.  the  maximum  pressure  of  ether  vapour  is  288  mm. ; 
that  of  alcohol,  24  mm. 

In  all  experiments,  therefore,  with  volatile  liquids  the 
vapour  pressure  must  be  allowed  for. 

For  example,  suppose  a  gas  is  collected  over  water  when 
the  temperature  is  20°  C.  Let  the  volume,  after  depressing  the 
jar  so  that  the  level  of  the  water  inside  the  jar  is  equal  to  the 
level  outside,  be  25  c.cm.,  and  let  the  corrected  barometer 
read  750  mm. 

The  pressure  inside  the  jar  equals  the  pressure  outside, 
which  equals  750  mm.  of 'mercury;  but  the  pressure  inside  is 


1 50  Heat 

made  up  of  the  pressure  of  the  gas  and  the  pressure  of  water 
vapour  at  20°  C. ;  this  equals  17-4  mm.  Therefore  in  any 
question  relating  to  the  gas  we  must  consider  it  collected  at 
20°  and  at  a  pressure  of  750  —  17-4  =  732-6  mm.  of  mercury. 
The  vapour  pressure  of  mercury  below  100°  is  generally 
negligible,  but  is  allowed  for  in  very  exact  experiments ;  at 
o°  C.  it  is  0*02  mm.,  at  100°,  0*746  mm. 

MAXIMUM  PRESSURE  OF  MERCURY  VAPOUR  IN  MILLIMETRES 
OF  MERCURY. 

o°.  0-02  mm.  80°.  o'35  mm.  160°.  5-9  mm. 

20°.  0-04     „  100°.  075     „  200°.  19-9     „ 

40°.  0-08     „  120°.  1-50     „  300°.  242-2     „ 

60°.  0-16     „ 

The  student  should  notice  that,  as  a  result  of  experiment, 
the  maximum  pressure  of  aqueous  vapour  at  100°  (i.e.  the 
boiling  point)  is  equal  to  the  pressure  of  the  atmosphere. 
Similar  experiments  show  that  the  maximum  pressures  of  all 
liquids  at  their  boiling  points  are  equal  to  the  atmospheric 
pressures  to  which  they  are  subjected. 

98.  Vaporization  in  the  Presence  of  other  Gases.— 
If  the  experiments  be  repeated,  with  the  difference  that  dry 
air  or  any  other  dry  gas  occupies  the  Torricellian  vacuum,  a 
similar  depression,  with  a  maximum  for  each  temperature,  is 
shown,  with  the  difference  that  the  vapour  takes  a  longer  time 
to  evaporate  and  exert  its  maximum  pressure. 

Dalton  enunciated  the  following  laws  : — 

(1)  The  maximum  pressure  of  a  saturated  vapour  depends 
only  upon  the  temperature,  and  is  not  affected  by  the  presence 
or  absence   of  any  other  vapour   that   does  not   chemically 
affect  it. 

(2)  The  pressure  exerted  by  a  mixture  of  gases  or  vapours 
is  equal  to  the  sum  of  the  pressures  exerted  by  the  individual 
gases  or  vapours. 

The  laws  seem  to  be  exact  enough  for  most  purposes,  and 
in  calculations  it  is  assumed  that  they  are  true,  as,  for  example, 
in  meteorology. 


Change  of  State —  Vaporization  and  Condensation    151 

99.  Boiling. — If  a  liquid  in  a  flask  or  beaker  be  heated  from 
below  by  a  lamp,  or  in  any  convenient  manner,  the  following 
changes  are  observed:  The  temperature  rises,  and  particles 
of  air  are  expelled;  there  is  also  an  increase  in  volume. 
The  parts  immediately  above  the  lamp  are  the  more  heated ; 
they  expand,  become  less  dense  than  the  particles  of  equal 
volume  above  them,  and  therefore  rise,  others  flow  in  from 
the  sides,  and  a  circulation  ensues.  The  temperature  at  the 
bottom  of  the  vessel  is  higher  than  the  temperature  of  the 
upper  layers,  and  bubbles  of  pure  water  vapour  are  formed ; 
these  bubbles  rise  and  condense  in  the  cooler  layers  above. 
The  condensations  follow  each  other  rapidly,  and  probably 
cause  the  "singing"  or  "simmering."  General  evaporation 
goes  on  from  the  surface  throughout  the  process,  increasing 
as  the  temperature  rises. 

Soon  the  general  temperature  is  such  (the  temperature 
has  risen  throughout  the  process)  that  the  bubbles  of  pure 
vapour  are  no  longer  condensed,  and  escape  from  the  free 
surface.  These  are  formed  rapidly,  and  ebullition  begins. 
The  temperature  in  the  case  of  water  is  now  about  100°  C, 
but  as  soon  as  ebullition  begins,  no  further  rise  takes  place. 

Boiling,  unlike  evaporation,  is  not  confined  to  the  free 
surface  of  the  liquid.  The  change  and  interchange  of  mole^ 
cules  of  liquid  and  vapour  suggested  in  evaporation  no  longer 
take  place. 

In  boiling,  the  molecules  of  vapour  just  formed  chase  the 
others  away,  driving  out  also  the  particles  of  air.  The  vessel 
above  the  liquid  is  now  filled  with  molecules  of  water  vapour, 
and  the  pressure  of  the  vapour  must  evidently  be  equal  to 
that  of  the  atmosphere  above  it,  which  it  supports.  We  thus 
deduce  the  following  laws  of  ebullition  : — 

(1)  The  boiling  point  of  any  liquid  is  a  temperature  that 
is  definite  for  a  given  pressure. 

(2)  The  temperature  of  the  boiling  point  remains  constant 
as  long  as  there  is  any  liquid  to  vaporize,  provided  the  pressure 
remains  constant 

(3)  The  pressure  of  the  vapour  of  a  liquid  during  free 
ebullition  is  equal  to  the  pressure  of  the  atmosphere  at  the  time. 


152 


Heat 


TABLE  OF  BOILING  POINTS  UNDER  NORMAL  ATMOSPHERIC 


PRESSURE  (C.°). 


-182° 

Petroleum 

...     106° 

_8o° 

Acetic  acid 

...        120° 

-34° 

Aniline      ... 

...     180° 

-10° 

Phosphorus 

...   290° 

35° 

Sulphuric  acid 

...  326° 

66° 

Mercury    ,  .  . 

...     350° 

78° 

Paraffin     ... 

...     370° 

100° 

Sulphur     .  .  . 

...     440° 

o 

104 

Zinc 

...     940° 

Oxygen  ... 
Carbon  dioxide 
Chlorine 

Sulphur  dioxide... 
Ether  ...  ""7.7 
Wood  spirit 

Alcohol 

Water     

Sea  water 


The  constancy  of  the  boiling  point  for  liquids  may  be 
readily  verified  by  measuring  the  temperature  of  ebullition. 

Even  pure  liquids,  however,  are  affected  by  the  nature  of 
the  vessel,  and  the  temperature  of  each  layer  is  not  always 

the  same.  In  determining  the 
boiling  point,  therefore,  the  tem- 
perature of  the  vapour  is  noted. 

Fig.  55  can  be  conveniently 
used  in  determining  boiling  points, 
instead  of  the  apparatus  shown  in 
Figs.  7  and  8.  A  small  hole,  a, 
is  made  in  a  test-tube.  The  test- 
tube  is  inserted  in  a  clean  flask 
containing  the  liquid.  A  bent 
tube,  £,  dipping  nearly  to  the 
bottom  of  the  tube,  allows  for 
the  escape  of  the  vapour  (it  should 
be  wider  than  shown ;  the  violent 
escape  of  steam  in  the  figure  sug- 
gests a  notable  increase  of  pres- 
sure inside  the  test-tube).  By 
following  the  arrows  it  will  be  seen 
that  the  vapour  in  the  test-tube 
is  kept  from  condensing  by  the  vapour  in  the  flask. 

The  reading  of  the  thermometer  will  give  the  temperature 


FIG.  55- 


Change  of  State —  Vaporization  and  Condensation    153 

of  the  boiling  point  for  the  pressure  of  the  atmosphere  at  that 
time,  as  given  by  the  standard  barometer. 

The  temperature  of  the  boiling  point  for  pressures  near  the 
normal  atmospheric  pressure  has  been  determined  with  great 
accuracy.  In  determining  the  upper  fixed  point  of  a  thermo- 
meter (§  4)  the  point  obtained  is  the  temperature  opposite 
the  barometric  pressure  in  the  following  table  at  the  time  of 
the  experiment.  For  example,  if  at  the  time  of  the  experiment 
the  corrected  barometric  height  was  754  mm.,  the  point  deter- 
mined would  be  99*8°  C. ;  if  the  barometer  stood  at  775,  the 
point  would  be  between  100-5°  and  100-6°  C. 

PRESSURE  OF  AQUEOUS  VAPOUR  BETWEEN  99°  AND  101°  C.1 

At  99°.  At  100°. 


mm. 


o'o 733*2  760-0 

o-i  73S'8  7627 

0-2  ...         ...  738-5  ...         765-5 

0-3  741-2  768-2 

o*4 •  743*8  ...  77i'9 

0-5  *..         ...  746-5  ...  7737 

0-6     749-2  ...  776-5 

07  ,..         ...  751-9  ...  779-3 

0-8     754*6  ...  782-0 

°*9     757*3  784*8 

If,  as  in  Fig.  17,  the  boiling  point  be  simply  determined, 
it  is  only  necessary  to  connect  the  two  tubes  so  that  the  steam 
has  to  overcome  the  pressure  of  a  certain  depth  of  water,  or, 
better,  of  a  certain  depth  of  mercury,  to  show  a  rise  in  the 
boiling  point  due  to  an  increase  of  pressure. 

100.  Effect  of  Pressure  upon  the  Boiling  Point. — 
Water  may  be  made  to  boil  at  any  temperature.  The  neces- 
sary and  sufficient  condition  is  that  the  pressure  should  be 
the  pressure  required  for  the  given  temperature  (see  Table, 

§97)- 

Illustrations  of  water  boiling  at  low  temperatures  are — 
(i)  Water  is  placed  in  a  vessel  under  the  receiver  of  an 
1  Kohlrausch's  physical  measurements. 


154 


Heat 


air-pump ;  the  vessel  rests  in  a  larger  dish  containing  strong 
sulphuric  acid.  The  air  is  exhausted,  the  pressure  reduced, 
and  evaporation  of  the  water  takes  place,  the  vapour  formed 
being  absorbed  by  the  sulphuric  acid.  When  the  pressure  is 
lowered  sufficiently  the  water  boils.  This  classical  experiment, 
clue  to  Leslie,  is  not  easy  to  perform.  The  sulphuric  acid 
does  not  absorb  readily,  and  the  vapour  spoils  the  tubes  and 
valves  connected  with  the  air-pump.  In  all  cases  it  is 
advisable  to  have  the  water  slightly  warmed. 

(2)  Another  form  of  the  same  experiment  is  to  boil  water 


FIG.  56. 


FIG.  57. 


briskly  in  a  flask ;  invert  it,  and  allow  it  to  cool.  On  pouring 
cold  water  over  it,  it  begins  to  boil  (Fig.  56). 

The  cold  water  condenses  part  of  the  vapour,  and  reduces 
the  pressure  inside  the  flask  sufficiently  to  produce  ebullition. 
Pouring  hot  water  over  it  stops  the  boiling. 

101.  Papin's  Digester. — Papin's  digester  (Fig.  57)  is  used 
for  boiling  liquids  at  high  temperatures.  The  cover  of  the  strong 
metallic  vessel  M  is  fastened  down  by  a  screw.  The  lever  l>9 


Change  of  State —  Vaporization  and  Condensation    155 

whose  fulcrum  is  at  a,  presses  upon  a  rod,  u,  whose  base  is  a 
valve  pressing  upon  a  hole  in  the  cover.  As  the  pressure 
increases  it  raises  u,  and  the  steam  escapes.  The  pressure  is 
regulated  at  from  5  to  6  atmospheres  by  the  weight  /,  the 
amount  is  read  from  the  graduated  bar  d\  thus  the  boiling 
point  will  be  between  150°  and  160°  C.  At  this  temperature 
the  gelatine  can  be  extracted  from  bones.  At  high  altitudes 
a  modification  can  be  used  for  cooking. 

102.  Hypsometers. — As  we  ascend,  the  pressure  of  the 
atmosphere  diminishes,  and  therefore 

water  boils  at  a  lower  temperature 
than  100°  C.  At  (Quito  the  boiling 
point  is  90°;  on  the  top  of  Mont 
Blanc,  85-5°;— temperatures  insuf- 
ficient for  many  cooking  operations. 

Travellers  are  able  to  use  a 
hypsometer  as  a  convenient  way  of 
determining  altitudes.  It  consists  of 
a  copper  vessel  (Fig.  58)  supplied 
with  delicate  thermometers,  openly 
divided  between  80°  and  100°  C,  so 
that  the  tenths  of  a  degree  can  be 
readily  estimated.  The  vessel  is  tele- 
scopic, and,  when  closed,  occupies 
small  space. 

Roughly,  a  diminution  of  i°  C. 
in  the  boiling  point  represents  an 
increase  in  altitude  of  1080  feet. 
For  travellers,  properly  calculated 
tables  are  constructed ;  its  porta- 
bility gives  it  a  great  advantage  over 
barometers,  and  clean  snow  is  gene- 
rally available. 

103.  Saline  Solutions. — Salts  dissolved  in  water  raise  the 
boiling  point.    The  boiling  point  increases  as  the  salt  is  added, 
until  the  solution  is  saturated  and  is  unable  to  dissolve  more. 

In  fixing  the  boiling  point  in  the  thermometer  pure  water 
was  used.     It  was  at  one  time  thought  that  this  was  not  neces- 


156  Heat 

sary,  as  the  steam  from  a  saline  solution  was  said  to  be  at  the 
temperature  of  boiling  water  boiling  under  the  same  pressure ; 
it  seems,  however,  proved  that  it  is  higher. 

The  boiling  point  of  a  liquid  is  affected  by  the  nature  of 
the  vessel ;  water,  for  example,  boils  at  a  higher  temperature  in 
glass  than  in  a  copper  vessel,  the  difference  being  about  i°.  If 
the  glass  vessel  be  thoroughly  cleaned,  there  seems  a  difficulty 
in  boiling  beginning,  and  sometimes  the  temperature  will  rise 
to  105°  or  106°  C.,  when  it  suddenly  begins  with  "  bumping." 
Boiling  with  "  bumping  "  is  avoided  by  adding  pieces  of  metal 
(platinum)  to  the  liquid. 

Water  freed  from  air  by  boiling  and  then  allowed  to  cool 
can  frequently  be  raised  by  reheating  above  100°,  in  some  cases 
up  to  130°,  before  signs  of  boiling  are  shown,  but,  on  boiling, 
the  liquid  is  suddenly  changed  into  vapour  with  explosive  force. 

104.  Latent  Heat  of  Vaporization. — During  the  time 
that  boiling  takes  place,  the  temperature  remains  constant.  The 
heat-energy  is  utilized  in  separating  the  molecules,  so  that  the 
substance  becomes  a  vapour.  This  energy  exists  as  potential 
energy,  and  when  the  molecules  close  up  in  condensation,  the 
heat-energy  again  appears  as  sensible  heat.  In  the  change 
from  liquid  to  vapour  there  is  a  great  increase  in  volume ;  thus 
i  cubic  inch  of  water  at  100°  becomes  1700  cubic  inches  of 
steam  at  100°.  The  vapour,  therefore,  if  surrounded  by  air, 
say,  has  to  force  the  air  back,  and  in  so  doing  does  external 
work.  The  internal  work  is  represented  by  overcoming  the 
cohesion  of  the  molecules,  and  in  certain  work,  such  as  in- 
creasing the  rotation,  etc.,  of  the  molecules.  This  being  done, 
if  the  source  of  heat  be  removed,  the  surrounding  air  presses 
upon  the  vapour  and  does  work  upon  it,  the  molecules  come 
closer  together,  the  potential  energy  is  liberated,  and  appears 
then  as  sensible  heat 

The  heat  used  during  vaporization  was  said  to  be  "  latent ;  " 
the  real  meaning  can  now  be  understood. 

The  number  of  units  of  heat  required  to  change  one  unit 
of  mass  of  a  liquid  at  its  boiling  point  into  vapour  without 
raising  its  temperature,  is  called  the  latent  heat  of  vaporization 
of  the  liquid. 


Change  of  State —  Vaporization  and  Condensation     1 57 

105.  Dr.*  Black's  Method. — Dr.  Black  made  one  of  the 
first  determinations  by  applying  heat  to  a  small  quantity  of 
water  at  50°  F.  (suppose  the  mass  to  be  i  oz.),  keeping  the 
supplied  heat  as  regular  as  possible.  He  noted  the  time  that 
elapsed  before  the  liquid  boiled ;  still  supplying  heat,  he  noted 
how  long  it  took  to  evaporate  the  water, — this  latter  operation 
took  five  times  longer  than  it  did  to  raise  the  water  to  boiling 
point. 

He  then  argued  that  to  raise  a  given  mass  from  50°  to  212° 
requires  162  units  (unit  of  mass  i  oz.,  degrees  Fahrenheit) ; 
therefore  to  evaporate  the  mass  takes  162  x  5  =  810  units. 
The  method  cannot  give  accurate  results.  No  note  is  taken 
of  the  loss  due  to  radiation  and  the  loss  of  water  due  to 
evaporation,  and  his  source  of  heat  would  vary.  The  method 
is  instructive,  and  analogous  to  that  followed  in  determining 
the  latent  heat  of  fusion  (§  82). 

He  next  evaporated  i  Ib.  of  water  at  its  boiling  point,  and 
passed  the  steam  formed  through  a  spiral  surrounded  by  40  Ibs. 
of  water  in  a  tube.  The  steam,  in  condensing,  raised  the 
temperature  20°,  and  he  inferred  that  the  latent  heat  of  steam 
was  800.  The  result  is  again  too  low;  but  this  method, 
modified,  is  still  used  in  the  methods  of  Despretz. 

106.  Modern  Method, — The  general  plan  followed  at 
present  is  to  allow  the  steam  to  pass  directly  into  water ;  it 
is  there  condensed,  and  raises  the  temperature  of  the  water. 
A  simple  form  of  the  experiment  can  be  made,  avoiding  small 
corrections,  as  follows  : — 

The  flask  A  (Fig.  59)  contains  300  grams  of  water;  the 
temperature  is  13°  C.  Water  is  boiled  in  C.  The  vapour 
passes  along  B  and  issues  from  D.  When  B  and  D  are  heated 
by  the  steam,  and  it  issues  freely,  D  is  dipped  into  A.  The 
steam  condenses,  heats  the  water  in  A.  In  four  to  five  minutes 
the  temperature  rises  to  52°  C.  D  is  removed,  and  A  is 
weighed.  The  increase  is  found  to  be  20*2  grams.  A  bright 
sheet  of  tin  prevents  radiation  between  the  two  flasks,  and  B 
catches  condensed  water  and  prevents  it  passing  along  D. 

(i)  300  grams  are  heated  through  39  degrees.  This 
requires  300  X  39  =  11,700  thermal  units. 


158  Heat 

(2)  This  is  supplied  by  (a)  20*2  grams  of  steam  at  100° 
passing   into   liquid  at  100° ;   and  (b)   20*2   grams  of  water 


FIG.  59. 

cooling  from  100°  to  52°.       (b)  gives  up  20-2  x  48  =  969-6 
thermal  units. 

11,700  —  969-6  =  10,730-4  thermal  units  must  be  supplied 
by  20 -3  grams  of  steam  condensing. 

.*.  i  gram  of  steam  condensing  gives  up     °'73°  4-.  ^34 
thermal  units 

The  latent  heat  of  evaporation  is  534  (in  accurate  experi- 
ments it  is  537).  There  are  many  obvious  causes  of  error,  due 
to  radiation,  to  water  passing  over,  and  to  water  condensing  in  D. 

Precaution  must  be  taken  to  see  that  the  steam  in  D  is 
not  cooled.  It  should  pass  through  a  copper  spiral  enclosed 
in  a  steam-jacket;  the  steam  in  the  spiral  will  thus  be  kept 
at  100°  C.  The  calorimeter  should  be  similar  to  that  shown 
in  Fig.  41,  but  larger,  its  water  equivalent  being  known. 

Let  M  =  mass  of  water  in  the  calorimeter ;   CD  its  water 

equivalent. 

/  =  initial  temperature  of  the  calorimeter. 
0  =  final  „  „  „ 

m  -  mass  of  steam  condensed. 
L  =  latent  heat  of  steam. 


Change  of  State  —  Vaporization  and  Condensation     1  59 

(1)  The  mass  of  steam,  w,  at  100°  condenses  to  water  at 
100°,  and  liberates  mL  units  of  heat 

(2)  m  units  of  water  cool   from  100°  to  0,  and   liberate 
w(ioo  —  6)  units  of  heat. 

(3)  The  water  in  the  calorimeter  and  the  calorimeter  gain 
(M  4-  <•>)(#  -  t)  units  of  heat. 

.'.  mL,  +  m(ioo  -  0)  =  (M  +  o>)(0  -  /) 


If  the  temperature  of  the  steam  be  T  instead  of  100°  (for 
example,  if  the  barometer  stood  at  750  mm.,  it  would  be  99*63° 
by  §  99),  then  the  exact  value  must  be  substituted  for  TOO 
in  the  above  equation. 

107.  Total  Heat  of  Steam.—  Regnault  made  a  large 
number  of  experiments  in  order  to  determine  the  total  heat 
necessary  to  raise  i  unit  of  mass  of  water  at  freezing  point  to 
any  given  temperature,  and  to  evaporate  it  at  that  temperature. 
This  is  called  the  "  total  heat  of  steam."  If  the  given  tempera- 
ture be  100°  C.,  then  the  two  quantities  are  roughly  100  +  537 
-  637  thermal  units. 

Regnault  concluded  that  the  total  heat  could  be  obtained 
at  any  temperature  from  the  formula  — 

Total  heat  =  606-5  -f-  o'3o5/ 

where  t  is  the  temperature  in  degrees  Centigrade.  The  total 
heat  increases  with  the  temperature. 

At  100°  the  total  heat  is  606-5  +  3^*5  =  637. 

If  the  experiment  be  conducted  so  that  the  water  is  all 
evaporated  at  50°  (that  is,  the  pressure  must  be  91*98  mm.)  — 

The  total  heat  =  606-5  +  I5'25  =  62175  thermal  units 

The  total  heat  of  steam  is  given  in  tables  for  a  wide  range 
of  temperature,  and  therefore,  knowing  the  heat  required 
to  raise  unit  of  mass  from  o°  C.  to  the  given  temperature 
(t.e.  determine  the  specific  heat  between  the  two  temperatures), 
the  latent  heat  of  evaporation  can  be  calculated  at  any  tem- 
perature. 


160  Heat 


Total  heat  }       (  Specific  heat 

I 

,  i 

'  Latent  heat 

of 

at/0 

5  ~: 

[  from  o° 

to/0 

j 

+  \ 

[  vaporization 

at/0 

T 

= 

S 

+ 

L 

>       T 

as 

606- 

5 

+ 

0*305^ 

also 

/.  L  =  T  -  S  =  606-5  +  0-305^  - 

Assuming  that  the  average  specific  heat  between  o°  and  f 
is  unity  — 

.-.   L  =  606-5  —  0-695^ 

The  latent  heat  at  o°  =  606-5 

10°  =  599*55 
„        „          100°  =  537 

The  latent  heat  diminishes  as  the  temperature  increases. 
We  can  calculate  the  temperature  at  which  the  latent  heat 
is  zero,  by  making  L  equal  to  zero  ;  then  — 

o  =  606*5  —  0-695^ 


This  temperature  is  far  above  that  of  any  possible  experiment 
with  steam,  and  we  cannot,  therefore,  attach  any  particular 
value  to  it.  It  suggests,  however,  that  at  some  particular 
temperature  the  liquid  will  pass  instantaneously  into  vapour. 
The  student  should  refer  to  this  after  §  117. 

The  student  will  find  no  difficulty  in  understanding  how 
vaporization  takes  place  at  any  temperature.  At  100°  the 
pressure  will  be  760  mm.  of  mercury;  at  50°  the  pressure  will 
be  92  mm.;  at  150°  C.  the  pressure  will  be  5  atmospheres; 
and  at  o°  the  pressure  will  be  4/6  mm. 

The  particulars  relating  to  water  in  its  three  states  can  now 
be  conveniently  collected. 

Temperature  of 
change  under  con. 
Specific  heat.  Latent  heat.         slant  pressure. 

(1)  Solid:  ice       ...       0-504      ...        80       ...       o° 

(from  -20°  to  o°  C.) 

(2)  Liquid  :  water         i  ...      537       ...    100° 

(3)  Vapour  :  steam        0-4805 


Change  of  State  —  Vaporization  and  Condensation     161 


Total  heat  of  steam  =  606*5  + 
Latent  heat  at  f  =  606-5  ~ 

108.  Latent  Heat  of  other  Liquids.  —  The.  following 
table  gives  the  latent  heats  at  normal  pressure  and  at  their 
ordinary  boiling  points.     The  latent  heat  of  steam  is  very  large 
compared  with  the  other  liquids,  and  explains  the  heat  and 
time  necessary  to  change  water  into  its  vapour  compared  with 
the  heat  and  time  required  to  change  any  other  liquid.     In 
nature  the  evaporation  (the  latent  heat  below  100°  is  greater 
than  at  100°)  of  water  takes  place  slowly,  and  when  water  vapour 
condenses  to  water  there  is  a  great  liberation  of  heat-units, 
explaining  the  rise  in  temperature  which  follows  rain,  and  the 
severe  scalds  from  steam. 

Latent  heat  of 
Boiling  point.  vaporization. 

Water      .........  100°  537 

Alcohol               ...         ...  78°  ...         202 

Sulphur  dioxide              ../  —  10°  ...           97 

Carbon  disulphide  48°  87 

Ether       ...                     ...  38°  80 

Turpentine          ...         ...  159°  ...           74 

Mercury              ......  350°  62 

Chloroform          ......  85°  ...           61 

109.  Coldness  due    to    Evaporation.—  If   the    heat 
necessary  for  evaporation  of  a  substance  be  not  supplied  by  an 
external  source  of  heat,  it  will  be  taken  from  the  substance  itself 
or  the  bodies  near  it,  and  the  temperature  will  fall.     To  this 
is  due  the  coldness  felt  when  water,  alcohol,  ether,  or  other 
volatile  liquids  evaporate  from  the  hand. 

The  temperature  of  a  room  is  lowered  by  sprinkling  water 
on  the  floor,  the  heat  required  for  evaporation  being  taken 
from  the  temperature  of  the  air.  Porous  water-vessels  keep 
the  water  they  contain  cool  by  the  evaporation  from  the 
outside  surface.  The  evaporation  of  the  perspiration  from  the 
skin  keeps  the  body  cool,  and  if  a  volatile  liquid  such  as  eau- 
de-Cologne  be  used,  the  reduction  of  temperature  takes  place 
more  rapidly;  its  latent  heat  of  evaporation  is  not  so  great, 

M 


1  62  Heat 

but  at  the  ordinary  temperature  it  is  nearer  its  boiling  point 
than  water,  and  evaporation  takes  place  the  more  readily. 
The  heat  liberated  in  condensation  is  utilized  in  manufac- 
tories; liquids  are  readily  heated  by  passing  steam  through 
them. 

110.  Internal  and  External  Work  in  Vaporiza- 
tion. —  The  energy  given  to  the  liquid  has  —  . 

(1)  To  do  internal  work  in  overcoming  the  cohesion  of  the 
molecules. 

(2)  To  do  external  work  in  expanding  the  liquid  against 
the  pressure  of  the  atmosphere. 

This  has  already  been  simply  discussed  in  §  78.  It  will 
form  a  good  exercise  to  do  it  here  with  greater  accuracy. 

i  volume  of  water  at  4°  C.  =  i-oooi  at  o°  C. 
=  1*0008  at  15°  =  1*0432  at  100°  C. 
=  1696  volumes  of  steam  at  100°  C 

the  pressure  throughout  being  that  of  the  normal  atmosphere 
=  14*7  Ibs.  on  the  square  inch. 

.*.   i  volume  of  water  at  100°  C.  =  1624  volumes  of  steam  at 
100°  C. 

To  vaporize  i  Ib.  of  water  at  100°  C.  takes  537  heat-units 
=  537  *  I39°  =  746543°  foot-pounds  of  energy. 
i  cubic  foot  of  water  at  100°  C.  weighs  59*64  Ibs. 

.*.  i  Ib.  of  water  will  represent  29  cubic  inches 

If  now  we  imagine  the  i  Ib.  of  water  in  a  cylinder  whose 
section  is  29  square  inches,  it  will  stand  at  i  inch  in  the  tube, 
and  when  vaporized  will  stand  at  1624  inches. 

The  pressure  on  the  water  is  14*7  x  29  =  426*3  Ibs. 


/.  external  work  done  =  426*3  x    -     foot-pounds 

12 

=  57,665*4  foot-pounds 
.*.  internal  work  =  746,430  —  57,665*4 

=  688,764*6  foot-pounds 
external  work  :  internal  work  ::  57,665*4  :  688,764*6 


Change  of  State —  Vaporization  and  Condensation    1 63 

111.  Ice-Machines. — In  Leslie's  experiment  (§  100),  if  the 
exhaustion  be  continued  far  enough,  and  the  sulphuric  acid 
absorbs  the  moisture,  the  liquid  not  only  boils,  but  its  tem- 
perature falls  so  that  it  freezes.  A  practical  modification  of 
this  is  Carre's  ice-machine  (Fig.  60). 

A  strong  cylinder,  R,  contains  strong  sulphuric  acid  intro- 
duced by  the  funnel  E.  P  is  an  air-pump  that  exhausts  air 


FIG.  60. 

from  the  cylinder  by  the  pipe  b,  connected  with  R  by  the 
tubulure  m.  The  dome  d  above  m  supports  obstacles  to 
prevent  the  spurting  of  acid  into  m  and  b.  The  pump  is 
worked  by  a  lever,  M,  which  at  the  same  time  moves  a  rod, 
/,  that  keeps  in  motion  a  stirrer,  A,  in  the  liquid  (the 
mechanism  is  shown  at  A,  the  intermediate  parts  being  x,  n, 
and  e).  A  wide  tube,  «,  is  covered  by  a  glass  disc,  O,  and 
through  the  glass  the  movement  of  the  acid  can  be  observed. 
The  interior  of  the  cylinder  is  connected  with  the  bottle 


1 64 


Heat 


containing  the  water  to  be  frozen,  by  a  tube,  h,  in  which 
is  inserted  a  stop-cock,  r.  On  pumping,  the  pressure  is 
reduced,  and  the  water  in  C  evaporates.  It  is  the  more  readily 
absorbed,  seeing  that  the  acid  is  stirred.  Soon  the  water 
boils.  When  this  stage  is  reached,  a  few  strokes  at  intervals 
of  a  few  minutes  continue  the  evaporation,  and  the  water 
freezes. 

The  principle  is  equally  well  illustrated  in  the  following 
experiment :  A  thin  copper  capsule,  C  (Fig.  61),  is  rested  on 
some  water  spilt  upon  a  board,  B.  Carbon  disulphide  is 
placed  in  the  capsule,  and  evaporated  by  blowing  over  it  with 


FIG.  61. 


FIG.  62. 


a  pair  of  bellows,  N.  Its  temperature  falls  so  much  that  the 
water  beneath  the  capsule  freezes.  This  should  be  performed 
in  the  open  air. 

Wollaston's  cryophorus  (Fig.  62)  consists  of  two  bulbs 
connected  by  a  bent  tube,  B.  Water  is  placed  in  it  and  boiled 
briskly.  It  is  then  hermetically  sealed,  so  that  it  contains 
only  water  and  its  vapour. 

All  the  water  is  placed  in  A,  and  the  other  bulb  is  placed 
in  ice  or  some  suitable  freezing  mixture.  The  evaporation 
from  A  soon  reduces  the  temperature  so  far  that  dew,  and 
then  snow,  forms  on  the  outside ;  ultimately  the  water  in  A 
freezes. 

112.  Freezing  of  Mercury. — To  effect  this  a  temperature 
of  —40°  C.  must  be  obtained.  Liquid  sulphur  dioxide  boils 
at  — 10°  C.  If  the  bulb  of  a  small  thermometer  be  covered 


Change  of  State —  Vaporization  and  Condensation    165 

with  cotton  wool  which  is  moistened  with  sulphurous  acid, 
then,  if  it  be  placed  under  the  receiver  of  an  air-pump,  and 
the  pump  be  exhausted,  the  rapid  evaporation  lowers  the  tem- 
perature below  —  40°,  and  the  mercury  is  frozen. 

By  selecting  liquids  that  are  extremely  volatile,  very  low 
temperatures  can  be  produced.  The  application  of  this  will 
be  seen  in  the  liquefaction  of  gases. 

113.  Evaporation. — The  conditions  favourable  to  evapo- 
ration are — 

(1)  The  area  of  the  free  surface. — Advantage  is  taken  of 
this  in  evaporating  large  masses  of  liquid.     For  example,  salt- 
pans are  made  as   large  and   as   shallow  as    possible.      A 
similar  plan  is  followed  in  evaporating  salt  water  on  the  sea- 
coast,  • 

(2)  The  pressure. — Dalton  demonstrated  the  relation  by  sup- 
porting a  dish  of  water  at  45°  F.  over  a  dish  of  concentrated 
sulphurous  acid,  and  placing  all  under  the  receiver  of  an  air- 
pump.      He   allowed   evaporation  to  go  on  for  30  minutes, 
keeping  the  pressure  constant  for  each  experiment,  and  then, 
by  weighing,  he  calculated  the  mass  of  water  that  had  evapo- 
rated.    His  experiments  showed  that  the  loss  due  to  evapo- 
ration was  inversely  as  the  pressure :  when  the  pressure  was 
30-4  inches   of  mercury,    1-24   gram   evaporated;    when  the 
pressure  was  15*2  inches,  2-97  grams  evaporated. 

In  §  95  it  has  been  shown  that  when  evaporation  takes 
place  in  vacuo,  the  rate  is  almost  instantaneous. 

(3)  The  pressure  due  to  two  vapours  is  equal  to  the  sum  of 
the  pressures  of  the  vapours  when  acting  separately.     Evapo- 
ration takes  place  independently  of  the   presence   of  other 
vapours,  but  the  rate  is  slower  (§  98). 

If  a  vapour  evaporates  into  a  confined  space,  the  limit  will 
be  due  in  all  cases  to  the  pressure  of  its  own  vapour.  The 
greater  the  amount  of  vapour  already  in  the  space,  the  slower 
will  be  the  rate  of  evaporation ;  and  as  each  particle  evaporated 
increases  the  amount  in  the  space,  the  slower  will  be  the  next 
small  quantity  in  evaporating. 

We  see,  therefore,  that  any  circumstance  that  will  remove 
vapour  when  formed  will  assist  evaporation.  Evaporation  will 


1 66  Heat 

take  place  more  readily  in  dry  air  than  in  moist  air; 
it  will  be  assisted  by  winds  removing  the  moist  air; — facts 
noticeable  in  nature.  The  removal  of  the  moist  air  by  absorp- 
tion is  the  function  of  the  sulphuric  acid  in  Leslie's  and  Carry's 
machines. 

The  greater  the  pressure  of  the  vapour  of  the  liquid,  the 
more  readily  will  it  overcome  the  resistance  of  any  gas  or 
vapour  present,  and  the  more  rapid  will  be  the  evaporation. 
We  increase  the  pressure  by  raising  the  temperature,  and  it  is 
well  known  that  evaporation  is  assisted  by  raising  the  tempera- 
ture of  the  liquid  as  nearly  as  possible  to  the  boiling  point. 

The  whole  subject  will  be  further  illustrated  in  the  chapter 
on  climate. 

114.  Sublimation. — Reference  hat  already  been  directed 
to  the  fact  that  ice  disappears  from  the  streets  without  passing 
into  the   liquid   state;   at  any  rate,  we  are  unable  to  detect 
the  formation  of  liquid.     This  process  is  called  sublimation. 
It  can  be  illustrated  by  heating  iodine  or  ammonium  chloride 
in  a  test-tube.      The  gradual  disappearance  of  solid  camphor 
is  a  further  illustration. 

115.  Distillation. — Distillation  is  another  illustration  of 
evaporation  and  condensation.    Evaporation  is  produced  ordi- 
narily by  the  heat  from  a  lamp;   the  vapour  passes  from  a 
region  of  high  pressure  to  one  of  lower  pressure,  and  is  con- 
densed, the   condenser  being  kept   cold   by  running  water. 
Liebig's  is  a  well-known  form  of  laboratory  condenser  (Fig.  63). 

The  condenser  j  is  a  glass  tube  that  surrounds  another  tube, 
g;  the  ends  of  j  are  made  water-tight  by  rings  of  indiarubber 
tubing.  The  condenser  is  kept  cold  by  water  entering  at  the 
lower  end  after  passing  through  the  funnel  /and  the  tube  B, 
and  leaving  it  (warmed,  of  course,  since  the  latent  heat  of 
vaporization  becomes  sensible  in  condensation)  at  the  upper 
end  by  the  tube  /.  The  liquid  is  placed  in  a  retort,  A,  the 
stem  of  which  fits  tightly  into  g,  which,  in  its  turn,  fits  loosely 
into  the  receiver  C,  in  which  the  condensed  liquid  collects. 
In  some  apparatus  the  vapour  passes  through  a  worm  kept 
in  a  large  vessel  of  cold  water. 

For  many  purposes  (as  sugar-refining)  a  high  temperature 


Change  of  State —  Vaporization  and  Condensation    167 

would  be  fatal  to  success.  Distillation  is  then  attained  by 
reducing  the  pressure,  thereby  introducing  evaporation  at 
lower  temperatures  (§  100). 

116.  Fractional  Distillation. — If  a  number  of  volatile 
liquids  that  do  not  chemically  combine  be  mixed  together,  the 
substance  with  the  lower  boiling  point  distils  at  that  lower 


FIG.  63. 

temperature  at  a  greater  rate  than  the  others.  If  a 
mixture  of  alcohol  and  water  be  distilled  slowly,  the  first 
distillate  up  to  the  boiling  point  of  alcohol  (80°  C.)  contains 
the  greater  proportion  of  the  alcohol  If  this  be  again  dis- 
tilled, a  distillate  stronger  in  alcohol  is  obtained.  This  is 
termed  "fractional  distillation."  By  its  aid  liquids  with 
different  boiling  points  can  be  roughly  separated.  Seeing  that 
evaporation  of  liquids  goes  on  at  all  temperatures,  pure  sub- 
stances cannot  be  obtained  by  this  process  alone. 

WORKED  EXAMPLES. 

I.  The  specific  heat  of  mercury  is  0*03.  A  pound  of  steam  at  100°  C. 
is  made  to  pass  into  a  vessel  containing  300  Ibs.  of  mercury  initially  at 
o°  C.,  the  capacity  for  heat  of  the  vessel  being  equal  to  that  of  10  Ibs.  of 
water  :  what  will  be  the  temperature  of  the  vessel  and  the  contents  at  the 
end  of  the  experiment  ?  Latent  heat  of  steam  is  537.  (London  Matric.) 

Let  /  =  temperature. 

(1)  Water  loses  537  4-  (100  —  /)  heat-units. 

(2)  Mercury  and  vessel  gain  (300  X  o'O3)/  +  io/  heat-units. 


1 68  Heat 

.'.  537  +  ioo  -  t  =  gt  +  lot  =  igt 
.*.  20t  =  637 
/./  =  31-85°  C. 

2.  A  barometer  tube  is  filled  with  mercury  and  inverted  over  mercury. 
At  first  the  column  stands  at  754  mm.,  and  agrees  with  the  standard 
barometer.  The  vacuum  measures  50  mm.  Dry  air  is  forced  up  until 
the  column  measures  724  mm.,  and  then  water  is  forced  up  the  tube 
until  a  thin  layer  forms  ;  the  mercury  column  is  depressed  and  measures 
700*6  mm.  Calculate  the  maximum  pressure  of  aqueous  vapour  at  the 
temperature  of  the  room,  10°  C. 

(1)  The  dry  air  occupies  50  +  30  =  80  mm.  of  the  tube,  and  is  subject 
to  a  pressure  of  754  —  724  =  30  mm.  of  mercury. 

(2)  After  the  water  vaporizes,  the  volume  of  the  air  is  30  +  (724  —  700-6) 
=  SS^. 

.'.  pressure  =  ^ =  44-9 

53 '4 

(3)  The  pressure  of  the  aqueous  vapour  (x)  +  the  pressure  of  the  air  + 
the  pressure  of  700-6  mm.  of  mercury  balances  the  pressure  of  the  atmo- 
sphere =  754  mm. 

.'.  x  +  44-9  +  700-6  =  754 

.*.  x  =  8-5  mm. 

EXAMPLES.    VII. 

1.  How  is  "  latent  heat "  explained  by  the  dynamical  theory  of  heat? 

2.  Describe  the  cryophorus,  and  explain  the  principle  of  its  action. 

3.  If  the  latent  heat  of  vaporization  be  537  when  one  degree  Centigrade 
is  the  unit  of  temperature,  what  will  it  be  when  one  degree  Fahrenheit  is 
the  unit  ?   Would  your  result  be  different  if  the  unit  of  mass  were  changed  ? 

4.  If  30  grams  of  steam  at  100°  C.  be  passed  into  400  grams  of  ice- 
cold  water,  what  will  be  the  temperature  of  the  mixture,  assuming  the 
latent  heat  of  steam  to  be  537  ? 

5.  At  the  sea-level  the  barometer  stands  at  750  mm.,  and  the  tempera- 
ture 7°  C.,  while  on  the  top  of  a  mountain  the  barometer  stands  at  400 
mm.,  and  the  temperature  —13°  C. :  compare  the  weights  of  a  cubic  metre 
of  air  in  the  two  places. 

6.  How  would  you  proceed  to  determine  the  maximum  pressure  of  the 
vapour  of  a  mixture  of  sulphuric  acid  and  water  at  various  temperatures  ? 

7.  Find  the  temperature  after  mixing  (a)  2  Ibs.  of  ice  at  o°  C.  with 
7  Ibs.  of  water  at  50°  ;  (3)  4  Ibs.  of  ice  at  — 10°  C.  with  30  Ibs.  of  water  at 
20°,  and  passing  I  Ib.  of  steam  into  the  mixture. 

8.  Find  the  total  heat  of  steam  at  60°  and  at  100°  C. 

9.  How  much  steam  must  be  approximately  passed  into  4  gallons  of 
water  at  60°  F.  to  raise  the  temperature  to  180°  F.  ? 

10.  Steam  is  passed  through  a  copper  worm  placed  in  a  calorimeter. 


Change  of  State —  Vaporization  and  Condensation    169 

The  worm  weighs  20*5  grams ;  its  specific  heat  is  0*095.  The  calorimeter 
is  also  of  copper,,  and  weighs  40  grams ;  it  contains  250  grams  of  water, 
and  the  initial  temperature  is  12°  C.  The  temperature  of  the  steam  is 
100°  C.  3*82  grams  of  steam  are  condensed,  and  the  condensed  water 
leaves  the  worm  at  a  temperature  of  20°  C.,  being  the  final  temperature 
of  the  calorimeter.  Calculate  the  latent  heat  of  steam. 

n.  Describe  Regnault's  method  of  determining  the  maximum  tempera- 
ture of  water  vapour.  300  c.cm.  of  hydrogen  are  collected  over  water. 
The  corrected  barometer  reads  750  mm.,  and  the  temperature  is  20°  C.  The 
water  is  at  the  same  level  inside  and  outside  the  tube.  Calculate  the  volume 
of  dry  hydrogen  at  normal  temperature  and  pressure. 

12.  What  effect  has  pressure  upon  the  boiling  point  of  a  liquid  ?     How 
would  you  experimentally  illustrate  your  answer  ? 

13.  A  steam  heating  apparatus  is  worked  with  a  boiler  that  indicates, 
a  pressure  of  20  Ibs.  on  the  square   inch  :    find  the  temperature  of  the 
steam  in  the  boiler.     What  will  it  be  when  the  pressure  rises  to  30  and  40 
Ibs.  per  square  inch  ? 


170  Heat 


CHAPTER  VIII. 

LIQUEFACTION  OF  GASES— VAPOUR-DENSITIES. 

117.  Isothermals  for  Steam. — An  isothermal  for  air 
(§  42)  forms  for  all  practicable  pressures  a  continuous  curve. 
In  the  case  of  steam  or  any  vapour  that  can  be  condensed  we 
obtain  an  isothermal  that  is  not  continuous.  Let  us  examine 
the  case  of  steam  at  100°  C.  If  the  pressure  at  the  beginning 
be  considerably  below  i  atmosphere — say  a  pressure  equal  to 
a  column  of  mercury  200  mm.  in  height — then,  if  we  slowly 
increase  the  pressure,  keeping  the  temperature  constant,  and 
record  the  results  graphically,  we  form  at  first  an  isothermal 
that  is  nearly  a  rectangular  hyperbola.  As  the  pressure  is 
increased,  the  curve,  instead  of  continuing  to  agree  with  the 
hyperbolic  curve,  falls  below  it ;  that  is,  the  volume  is  less  than 
it  should  be  if  calculated  according  to  Boyle's  law.  This 
decrease  in  volume  continues  as  we  approach  a  pressure  of 
760  mm.  When  the  pressure  is  760  mm.,  or  i  atmosphere, 
the  steam  is  on  the  point  of  being  condensed.  The  short 
limb  of  the  Boyle  tube  (Fig.  33)  could  be  surrounded  by  a 
steam-jacket  kept  at  100°  C. ;  the  mercury  in  both  limbs  would 
be  level  when  the  pressure  is  i  atmosphere  (§  100).  If  now 
we  attempt  to  increase  the  pressure  by  pouring  mercury  into 
the  long  limb,  we  observe  that  the  mercury  in  both  limbs 
continues  level,  showing  that,  although  the  volume  of  steam 
is  decreasing,  the  pressure  is  keeping  constant  at  i  atmosphere. 
The  other  observable  effect  is  that  part  of  the  vapour  is 
condensing,  and  a  layer  of  water  is  formed.  The  isothermal 
up  to  the  point  when  condensation  begins  is  the  curve  a  I 


Liquefaction  of  Gases — Vapour-Densities        171 
(Fig.  64).     With  diminished  volume  we  have  constant  pressure 


293. 


29V 


275 


266' 


257° 


248 


239° 


, 


30  40 

Cubic  feet. 
FIG.  64. 


represented  by  the  horizontal  line  b  c.     This  simply  represents 


172 


Heat 


the  fact  that  at  100  C.  (212°  F.)  the  maximum  pressure  of 
water  vapour  is  i  atmosphere,  and  no  higher  pressure  is 
possible  as  long  as  the  temperature  is  212°  F. 

In  the  diagram  the  isothermals  are  for  i  Ib.  of  steam. 
Volume  is  measured  in  cubic  feet,  and  pressure  in  atmospheres. 
When  the  volume  becomes  0*016  cubic  foot  (too  small  to  be 
represented  on  the  figure),  the  whole  of  the  steam  is  condensed ; 
and  to  continue  the  curve  would  give  the  isothermal  of  water 
at  2T2°F.  It  requires  enormous  pressure  to  produce  any 
appreciable  diminution  of  volume ;  so  that  the  curve  from  c 
would  be  almost  vertical.  If  the  isothermal  for  302°  F.  be  drawn, 
we  find  from  the  tables  that  the  maximum  pressure  of  water 
vapour  at  that  temperature  is  69-2 7  Ibs.  on  the,  square  inch, 

equal  to =47  atmospheres.     The  isothermal  up  to  this 

14*70 

pressure  is  d  e,  the  volume  of  steam  being  6-153  cubic  feet. 
The  pressure  now  remains  constant  until  the  whole  is  condensed 
into  water  at  302°  F.,  when  the  volume  is  0*0176  cubic  foot. 

If  we  took  the  temperature  at  356°  F.,  the  volume  at  the 
point  where  condensation  begins  would  be  2*025  cubic  feet, 
the  pressure  being  10  atmospheres.  When  condensation  is 
completed,  the  volume  is  o'oiS  cubic  foot  of  water. 


i  Ib.  of  steam. 

Maximum 

Volume  in  cubic  feet. 

Temperature 

pressure  in  Ibs. 

F. 

per 
square  inch. 

(a)  When  at  maximum 

(£)  Of  water  formed. 

212° 

1470 

26-360 

0-0l67 

302° 

69-27 

6-153 

0*0174 

392° 

226*07 

2-025 

0-0183 

4OI° 

250-57 

1-838 

0-0184 

We  notice  that,  as  the  temperature  increases,  the  volume 
of  steam  before  condensation  is  approaching  the  volume  of 
water  formed,  suggesting  that  at  some  very  high  temperature, 
probably  beyond  experiment,  the  volume  of  steam  would  be 
so .  nearly  equal  to  the  water  formed  that  it  would  be  difficult 


Liquefaction  of  Gases — Vapour- Densities        173 

to  detect  the  difference ;  that  is,  the  density  of  steam  before 
condensation  is  approaching  the  density  of  the  water  formed. 

The  latent  heat  of  steam  is  also  decreasing  (§107).  At  212° 
it  is  9657 ;  at  392°  it  is  838;  and  thus  a  smaller  quantity  of 
heat  is  necessary  to  effect  the  change  from  liquid  to  vapour; 
— liquid  and  vapour  are  approaching  a  state  of  unstable 
equilibrium.  It  will  be  seen  that  with  other  substances  we 
are  able  to  reach  a  state  when  this  unstable  equilibrium  is 
more  marked. 

Steam  at  pressures  very  much  lower  than  the  maximum 
pressure  at  which  it  can  condense  acts  like  a  perfect  gas,  and 
the  relation  PV  =  constant  is  approximately  true.  The  steam 
must,  however,  be  thoroughly  superheated. 

When  near  its  point  of  condensation  the  formula  which 
represents  the  relation  of  pressure  and  volume  is — 

17 

pyTF  _ 

a  formula  of  great  use  in  questions  relating  to  steam-engines. 

118.  Liquefaction  of  Gases. — Many  bodies  are  known  to 
us  both  in  the  liquid  and  gaseous  forms.  At  ordinary  pressures 
water  is  liquid  below  100°  C,  and  vapour  above  that  tempera- 
ture. Alcohol  is  liquid  below  80°  C.,  and  vapour  above  it 
Ether  is  liquid  at  35°  C.,  and  vapour  above.  Sulphur  dioxide, 
but  that  we  possess  means  of  obtaining  low  temperatures, 
would  only  be  known  as  a  vapour,  since  it  is  only  liquid  at 
temperatures  below  — 10°  C.  under  ordinary  atmospheric 
pressure;  and,  therefore,  in  some  Arctic  regions,  would  be 
generally  known  as  a  liquid  with  a  low  boiling  point. 

This  suggests  one  general  method  of  experimenting  upon 
gases  with  a  view  to  liquefying  them,  namely,  by  reducing  the 
temperatures. 

The  boiling  point  is  raised  by  increasing  the  pressure. 
Assuming  that  any  substance  ordinarily  known  to  us  as  a 
vapour  can  exist  as  a  liquid,  we  see  that  the  boiling  point 
must  be  at  some  temperature  lower  than  that  it  is  usually 
subjected  to.  But  increase  of  pressure  raises  the  boiling  point ; 
it  is  therefore  fair  to  infer  that,  if  we  increase  the  pressure  of 


Heat 

a  gas,  it  may  be  possible  to  so  raise  its  boiling  point  that  the 
gas  may  exist  as  a  liquid. 

The  two  methods — (i)  lowering  the  temperature,  (2)  in- 
creasing the  pressure,  or  combinations  of  both,  have  been 
the  methods  followed  in  liquefying  gases. 

The  liquefaction  of  water  vapour  can  be  readily  shown  by 
modifying  the  apparatus  of  Fig.  51.  The  amount  of  water 
formed  is  small,  and  will  be  the  more  readily  seen  if  the  upper 
part  of  the  tube  be  drawn  out  for  2  or  3  inches,  to  make  it  much 
narrower,  and  then  closed,  taking  care  to  keep  the  glass  thick 
enough  to  resist  the  pressure.  It  is  filled  completely  with  pure 
dry  mercury,  and  inverted  in  the  usual  way.  A  few  drops  of 
water  are  forced  up,  insufficient  to  form  a  layer ;  that  is,  so 
that  the  whole  vaporizes.  If  the  temperature  of  the  room  be 
20°  C.  say,  then  the  mercury  can  be  depressed  about  15  mm. 
(maximum  pressure  =  17*4  mm. ;  see  Table,  §  97)  without  a  layer 
of  water  forming.  The  pressure  will  be  approximately  15  mm. 
of  mercury.  Remove  the  tube  to  the  deep  trough  (Fig.  52), 
and  slowly  depress  the  tube ;  that  is,  increase  the  pressure  on 
the  vapour.  When  the  pressure  exceeds  18  mm.,  the  water 
will  begin  to  condense,  and  the  small  quantity  formed  can  be 
seen  in  the  narrow  part  of  the  tube.  By  further  depressing 
the  tube,  the  whole  can  be  liquefied.  If  at  the  beginning  a 
small  layer  forms,  it  can  be  vaporized  by  surrounding  the 
upper  part  of  the  tube  with  another  tube,  and  pouring  warm 
water  into  the  second  tube.  Suppose  the  water  is  at  60°  C., 
then  the  depression  produced  will  be  about  149  mm.  Still 
keeping  it  surrounded  with  the  warm  water  at  60°,  depress  it  in 
the  deep  trough,  and  the  water  vapour  will  be  condensed. 

By  raising  the  tube  rapidly,  the  boiling  of  the  liquid  under 
reduced  pressure  can  be  observed. 

Or  when  the  water  has  just  been  vaporized  by  the  heat 
from  the  warm  water,  if  the  warm  water  be  run  off,  condensa- 
tion (liquefaction)  takes  place  by  cooling,  and  the  mercury 
rises  in  the  tube. 

Ether  can  be  treated  similarly,  save  that  the  jacket  of  warm 
water  is  unnecessary,  and  that,  to  show  liquefaction  by  lowering 
the  temperature,  ice-cold  water  should  be  used. 


Liquefaction  of  Gases— Vapour-Densities        175 

Sulphur  dioxide,  as  made  in  the  laboratory,  is  a  gas.  If  it 
be  subjected  to  a  temperature  below  — 10°  C.  at  the  ordinary 
pressure,  it  becomes  liquid. 

The  gas  is  generated  in  the  flask  A  (Fig.  65),  and  is  passed 


FIG.  65. 

through  the  wash-bottle  B,  containing  strong  sulphuric  acid 
to  dry  it ;  it  then  flows  through  the  thin  glass  spiral  placed  in 
the  inverted  bell-jar  C.  C  con- 
tains a  freezing  mixture  of  ice  and 
salt,  or,  better,  ice  and  calcium 
chloride.  The  condensed  liquid 
sulphur  dioxide  is  collected  in  a 
vessel  surrounded  by  a  freezing  mix- 
ture of  ice  and  salt  contained  in  D. 
119.  Faraday's  Method.— 
Faraday  used  the  following  simple 
means  to  obtain  increased  pressure  : 
In  the  longer  limb  a  of  a  bent  tube  (Fig.  66)  is  placed  a 


FIG.  66. 


i/6  Heat 

substance  which,  either  by  heating  or  by  some  chemical  action, 
will  produce  the  gas  to  be  experimented  upon.  The  short 
limb  is  then  sealed.  On  heating  the  longer  limb  the  gas  is 
generated,  the  pressure  increases,  and  in  many  cases,  by  the 
increased  pressure,  liquefies,  in  the  short  limb.  In  some  cases 
the  short  limb  is  placed  in  a  freezing  mixture.  Usually  the 
amount  of  liquid  formed  is  small,  and  can  be  the  better  seen 
if  b  be  drawn  out  some  3  or  4  inches,  with  a  much  smaller 
diameter.  Care  must  be  taken  that  the  part  drawn  out  is 
strong,  and  that,  in  closing,  the  part  closed  is  properly  thickened. 

If  chlorine  gas  be  passed  into  water  just  above  the  freezing 
point,  crystals  of  chlorine  and  water,  called  hydrate  of  chlorine, 
and  containing  28^4  per  cent,  of  chlorine,  will  be  formed.  These 
crystals  are  rapidly  dried  with  blotting-paper,  and  placed  in 
the  limb  a,  which  is  then  inserted  in  a  freezing  mixture.  The 
limb  b  is  drawn  out  and  sealed,  b  is  now  placed  in  a  freezing 
mixture,  while  a  is  surrounded  with  warm  water.  The  hydrate 
decomposes,  and  forms  two  layers :  the  lower  is  liquid 
chlorine ;  the  upper,  water.  The  chlorine  distils  into  b,  and 
is  liquefied  by  the  increased  pressure. 

Silver  chloride  readily  absorbs  many  thousand  times  its 
volume  of  ammonia.  Dry  silver  chloride  (sufficient  to  nearly 
fill  the  limb)  is  placed  in  a  before  the  end  a  is  sealed.  The 
ends  a  and  b  are  drawn  out  ready  for  sealing.  Dry  ammonia 
is  now  passed  through  ;  it  is  absorbed  by  the  chloride  and  it 
also  drives  out  the  air.  Both  ends  are  sealed  b  is  placed  in 
a  freezing  mixture,  and  a  is  warmed  gently  with  the  Bunsen 
flame.  The  ammonia  absorbed  by  the  silver  chloride  is 
liberated,  and  the  pressure  is  sufficient  to  condense  it  in  b  as 
colourless  liquid  ammonia. 

The  simple  glass  tube  was  not  strong  enough  to  resist 
high  pressures;  it  was  replaced  by  iron  vessels,  the  pressure 
being  obtained  by  the  aid  of  force-pumps. 

By  such  methods  a  large  number  of  substances  that  had 
been  regarded  as  permanent  gases  were  liquefied,  and  proved 
to  be  vapours.  But  oxygen,  nitrogen,  hydrogen,  carbon 
monoxide,  and  marsh-gas  resisted  all  efforts,  and  were  still 
spoken  of  as  "  the  permanent  gases." 


Liquefaction  of  Gases — Vapour- Densities        177 

The  solution  of  the  problem  lay  in  the  researches  that 
follow. 

120.  Cagniard-Latour. — Cagniard-Latour  used  a  curved 
tube  closed  at  both  ends  (Fig.  67).     b  contained  air  above  the 
mercury,  and  served  as  a  pressure-gauge ;  a  contained  a  small 
quantity  of  alcohol.     Both  ends  were  graduated,  so  that  the 
volume  of  the  liquid  at  any  time  was  known,     a  was  placed  in 
a  bath,  and  heated  to  any  required  temperature,  and  the  follow- 
ing effects  were  noted  :  Part  of  the  alcohol 

vaporized,  but  on  increasing  the  temperature 
a  point  was  reached  where  no  distinction 
could  be  drawn  between  the  liquid  and  its 
vapour ;  the  volume  at  this  point  was  about 
three  times  its  original  volume.  The  sub- 
stance at  this  moment  was  at  a  temperature 
of  about  250°  C.,  and  the  pressure,  indicated 
by  £,  was  about  120  atmospheres. 

Drion  notices  that  the  meniscus  separating 
the  liquid  and  its  vapour,  which  at  first  was 
concave,  gradually  flattened  as  this  point  was 
reached,  until  it  became  a  plane;  then  the 
plane  of  demarcation  disappeared,  the  liquid 
and  the  vapour  gradually  approximated  to 
the  same  condition,  and  the  tube  was  filled 
with  a  vapour,  the  process  of  change  having 
been  gradual. 

This  suggests  that  at  a  sufficiently  high  temperature  the 
difference  between  a  liquid  and  a  vapour  no  longer  exists, 
and  that  the  liquefaction  of  the  gas  must  be  sought  for  under 
this  temperature.  (Compare  §  117.) 

121.  Andrews's  Researches. — Dr.  Andrews's  researches 
set  the  matter  in   a  clearer  light.      He  experimented   upon 
carbonic  acid  gas. 

The  gas  was  enclosed  in  a  strong  glass  tube,  g,  and  was 
shut  in  by  a  thread  of  mercury  (Fig. 68).  The  tube  was  firmly 
set  into  a  copper  tube,  E  R,  this  latter  being  filled  with  water. 
The  steel  screw  S  worked  into  this  tube ;  by  turning  it  the  required 
pressure  could  be  obtained.  The  pressure  was  measured  by 

N 


FIG.  67. 


178 


Heat 


connecting  the  interior  of  R  with  a  similar  tube,  in  which  was 
inserted  a  tube  similar  to  g,  but  containing  air.  The  second 
tube  formed  an  air-manometer,  and  was  previously  graduated 
to  measure  pressures  in  atmospheres.  The  glass 
tube  or  tubes  could  be  surrounded  by  a  glass  case 
containing  water,  which  could  be  kept  at  any 
required  temperature.  Care  was  taken  to  compress 
slowly,  and  allow  the  gas  to  cool  to  the  temperature 
required  by  the  experiment. 

The  results  were  graphically  set  out  (Fig.  69), 
the  ordinates  representing  pressure,  and  the  ab- 
scissae, volume.1  The  isothermals  for  various  tem- 
peratures were  drawn.  Take  the  case  when  the 
temperature  was  13-1°  C.  When  the  pressure  is 
about  48  atmospheres,  condensation  begins,  and 
the  liquid  and  gas  can  be  distinctly  seen;  the 
volume  of  the  liquid  after  condensation  is  com- 
plete is  about  one-fifth  of  the  volume  of  the  gas 
before  condensation  ;  until  condensation  is  com- 
pleted there  is  no  increase  in  pressure.  Liquid  car- 
bonic acid  being,  compared  with  its  gaseous  form, 
very  slightly  compressible,  little  diminution  of 
volume  is  shown  as  the  pressure  increases  to  about 
90  atmospheres, — the  isothermals  are  nearly  vertical. 
(Compare  this  with  the  isothermal  of  steam,  §  117.) 
Now  take  the  gas  at  a  temperature  of  21*5°  C. 
Liquefaction  only  begins  at  61  atmospheres.  The 
volume  of  the  liquid  obtained  when  all  is  condensed 
is  only  one-third  the  volume  of  the  gas  before  con- 
densation. 

There  has  been,  in  each  of  the  previous  experiments,  a 
distinct  break  between  the  liquid  and  gaseous  states.  When 
the  tube  is  at  a  temperature  of  31'!°,  the  curve  for  some  time 
suggests  the  isothermal  line  of  a  gas,  and  no  condensation 
ensues.  When  the  pressure  is  between  73  and  75  atmospheres, 
there  is  a  flattening  of  the  curve,  suggestive  of  the  flat  part 

1  In  Andrews's  paper  the  ordinates  represent  volume  and  the  abscissae 
pressure.     The  dotted  curve  is  also  absent 


FIG.  68. 


Liquefaction  of  Gases— Vapour-Densities        179 

of   the    13-1°   and    21-5°   lines,   but  no  physical  difference 
can  be  detected  in  the  gas.      When  the  pressure  increases 


FIG.  65. 


beyond  75,  the  diminution  of  volume  continues,  but  at  a  much 
slower  rate ;  it  is  suggestive  more  of  the  diminution  in  volume 


i8o 


Heat 


of  a  liquid.  The  same  points  are  observable  in  the  curves 
for  32*5°  and  35*5°,  but  they  disappear  for  48 'i°,  when 
carbonic  acid  seems  to  be  acting  like  a  perfect  gas,  and  obeys 
Boyle's  law. 

Dr.  Andrews  concluded  that  above  a  certain  temperature 
the  gas  cannot  exist  as  a  liquid ;  this  he  determined  to  be 
30*92°  C.  for  carbon  dioxide,  and  called  it  "  the  critical  tem- 
perature." He  states  that  other  gases  would,  like  carbonic 
acid,  possess  a  critical  temperature ;  in  fact,  carbonic  acid  gas, 
at  a  high  temperature,  behaves  like  one  of  the  so-called  per- 
manent gases. 

"  The  critical  temperature  of  what  are  called  the  permanent 
gases  is  probably  exceedingly  low,  so  that  we  cannot  by  any 
known  method  produce  a  degree  of  cold  sufficient,  even  when 
applied  with  enormous  pressure,  to 
condense  them  into  the  liquid  state." 
Since  this  sentence  from  Max- 
well's "Heat"  was  written,  it  has 
been  found  possible  to  cool  all  the 
permanent  gases,  save  hydrogen,  be- 
low their  critical  temperatures. 

The  distinction  between  gas  and 
vapour  then  disappears.  A  vapour 
is  a  gas  below  its  critical  temperature. 
Referring  again  to  Fig.  69,  we  see 
that  for  the  critical  temperature  there 
is  a  certain  pressure  where  condensa- 
tion begins  ;  it  is  somewhere  about  75. 
This  particular  pressure  is  called 
"the  critical  pressure."  When  it  is 
at  the  critical  temperature  and  the 
critical  pressure,  unit  mass  has  a 
definite  volume,  called  "the  critical 
volume." 

122.    Cailletet's     Method.— 
The  apparatus  (Fig.  70)  is  essentially 
The  capillary  tube  T  P  contains  the 


FIG.  jo. 

like  that  of  Dr.  Andrews. 


gas,  cut  off  by  mercury ;  the  lower  end  of  the  tube  is  curved 


Liquefaction  of  Gases — Vapour-Densities        181 

and  open,  and  dips  into  a  mercury  bath,  B.  The  tube  is  fitted 
into  the  bath  by  a  screw,  A.  Pressure  is  transmitted  from  a 
hydraulic  press,  not  shown,  by  means  of  the  tube  U  passing 
through  the  screw  RE.  T  P  is  surrounded  by  a  bath,  M, 
that  contains  a  freezing  mixture  when  necessary.  The  bath, 
in  turn,  is  surrounded  by  a  safety  bell-jar,  C. 

Carbonic  acid  gas  is  by  this  apparatus  easily  liquefied  at 
the  ordinary  temperature ;  but  oxygen  or  nitric  oxide,  even 
when  subjected  to  a  pressure  of  500  atmospheres,  shows  no 
signs  of  condensation.  If  after  compression  to  between  200 
and  300  atmospheres  the  heat  be  allowed  to  escape,  so  that 
the  gas  is  again  at  the  temperature  of  M,  and  the  screw  con- 
nected with  the  press  be  then  suddenly  opened,  so  that  the 
pressure  in  T  is  instantly  relieved,  a  mist  suddenly  appears 
and  rapidly  disappears ;  this  must  be  due  to  the  condensation 
of  the  special  gas  in  the  apparatus. 

We.  have  already  seen  that  heat  is  required  to  expand 
gases.  The  gas  expanding  when  the  great  pressure  is  relieved 
uses  heat-energy,  which  is  taken  from  itself.  There  is  thus 
a  sudden  fall  of  temperature  below  the  critical  temperature  of 
the  gas,  and  the  pressure  yet  remaining  is  sufficient  to  con- 
dense the  gas.  As  soon  as  the  pressure  is  further  relieved,  the 
liquid  evaporates.  By  this  apparatus  oxygen,  nitric  oxide,  and 
marsh-gas  were  liquefied. 

123.  Pictet's  Method. — At  the  same  period,  and  inde- 
pendently, Pictet  liquefied  these  "permanent"  gases  by  a 
different  method.  He  obtained  his  pressure  following  Faraday's 
method;  that  is,  he  heated  in  a  closed  tube  some  solid  or 
compound  that  would  generate  the  gas  required.  For  oxygen 
he  used  potassium  chlorate. 

The  substances  are  passed  into  the  iron  retort  L  through  P, 
which  is  then  securely  closed.  The  gas  generated  in  L— the 
heat  of  the  lamp  O  being  used  if  necessary— collects  in  M  (Fig. 
71) ;  the  increase  in  pressure  is  indicated  by  the  manometer  R. 
The  tube  M  is  surrounded  by  a  freezing  mixture  contained  in  H, 
that  produces  a  temperature  below  the  critical  point  of  oxygen. 

This  low  temperature  is  produced  as  follows :  The  con- 
denser K  is  surrounded  by  a  cylinder,  C,  containing  liquid 


1 82 


Heat 


sulphur  dioxide ;  on  working  the  pumps  A  and  B,  the  liquid 
is  evaporated  (following  the  arrows),  and  is  forced  into  a 
condenser,  D,  where  it  is  cooled  by  water  and  liquefied 
under  a  pressure  of  3  atmospheres.  It  then  passes  by  the 
tube  d  to  the  tube  C.  Thus  the  sulphur  dioxide  is  restored 
as  liquid  sulphur  dioxide,  but  at  a  lower  temperature,  —70°  C. 
Into  K,  surrounded  by  this  low  temperature,  is  forced 
carbonic  acid  gas  ;  this  gas  at  —70°  C.  liquefies  and  is  forced 


into  H,  passing  along  h.  The  gas  supply  is  now  cut  off, 
and,  by  working  the  pumps  E  and  F,  the  liquid  carbonic 
acid  is  evaporated  so  rapidly  that  it  freezes,  and  its  tem- 
perature sinks  to  —130°  C.  At  this  temperature,  in  the  case 
of  oxygen,  when  the  manometer  R  indicates  a  pressure  of 
500  atmospheres,  it  suddenly  sinks  to  300  atmospheres,  showing 
that  part  of  the  oxygen  has  condensed.  On  opening  the  stop- 
cock N  the  liquid  issues  as  a  white  (?)  jet  and  immediately 
evaporates.  (See  Dewar's  description.) 

Wroblewski  and  Olzewski  modified   Cailletet's  apparatus 


Liquefaction  of  Gases — Vapour- Densities        183 

to  produce  yet  lower  temperatures.  The  tube  containing  the 
gas  was  bent  twice  at  right  angles  (compare  T,  Fig.  70),  and 
the  end  dipped  into  a  vessel  containing  also  a  hydrogen 
thermometer.  The  vessel,  otherwise  closed,  could  readily  be 
connected  with  a  powerful  pump  to  evaporate  any  liquid  in  it 
Liquid  ethylene  flowed  from  a  reservoir  through  a  spiral  sur- 
rounded by  a  mixture  of  solid  carbonic  acid  and  ether,  and 
entered  the  vessel  at  a  temperature  of  about  —  100°  C.  The 
ethylene  was  evaporated,  and  the  temperature  fell  to  — 136°. 

At  this  temperature  oxygen  liquefies  under  a  pressure  of  only 
20  atmospheres.  By  using  liquid  oxygen  instead  of  ethylene, 
and  evaporating,  a  temperature  of  —200°  is  reached.  Vessels 
containing  air  or  nitrogen  at  a  pressure  of  100  atmospheres  are 
prolonged  by  a  tube  of  glass  that  is  placed  in  the  liquid.  If 
the  pressure  be  relieved,  they  at  once  liquefy  and  remain 
liquid.  The  connection  with  the  vessel  is  broken,  and  on 
opening  them  to  the  air  they  evaporate,  and  the  temperature 
rises  to  — 192°  C.  for  air  and  —  194°  for  nitrogen. 

Dewar,  by  improved  methods,  has  recently  been  able  to 
obtain  liquid  oxygen  in  comparatively  large  quantities.  It 
is  of  a  beautiful  sky-blue  colour,  possesses  marked  magnetic 
properties,  and  a  characteristic  spectrum;  it  boils  at  — 182°  C. 
under  ordinary  atmospheric  pressure,  and  therefore  rapid 
ebullition  takes  place  when  it  is  exposed  to  the  air.  He 
has  shown  that,  when  the  vessel  containing  the  liquid  is 
placed  in  a  glass  receiver,  and  a  vacuum  more  perfect  than 
any  hitherto  attained  is  formed  by  condensing  the  mercury 
vapour  in  a  Torricellian  vacuum  (the  vacuum  at  —  80°  C.  was 
about  0-000000003  of  an  atmosphere),  the  convection  of  heat 
when  the  residual  vapour  is  kept  still  is  greatly  reduced ;  the 
evaporation  of  the  liquid  is  so  far  lessened  that  it  is  possible 
to  carefully  examine  the  physical  properties  of  liquid  oxygen. 

By  evaporating  liquid  oxygen  in  a  suitable  receiver  (com- 
pare §  100),  a  temperature  is  reached  where  air  is  readily 
liquefied ;  by  a  similar  evaporation  of  liquid  air,  a  yet  lower 
temperature  is  obtained,  approaching  the  critical  temperature 
of  hydrogen,  which  gas  will  probably  soon  be  liquefied.  Air 
liquefies,  as  a  whole,  as  a  murky  blue  liquid,  but  in  boiling  it 


1 84  Heat 

follows  the  laws  of  mixed  liquids ;  nitrogen  distils  first,  then 
a  mixture  of  nitrogen  and  oxygen,  and  ultimately  almost  pure 
oxygen.  At  very  low  temperatures  liquid  oxygen  becomes 
inert  and  loses  its  ordinary  oxidizing  properties;  potassium, 
for  example,  that  tarnishes  at  once  in  air,  and  combines  with 
such  energy  with  the  oxygen  of  water  that  it  is  ignited,  floats 
unaltered  in  cooled  liquid  oxygen.  At  very  low  temperatures 
the  molecular  motion  is  so  far  reduced  that  the  properties  of 
matter,  as  at  present  understood,  seem  to  change. 

Boiling  points 

(i  atmosphere  pressure).    Critical  temperatures.     Critical  pressures. 
Degrees  Centigrade.       Degrees  Centigrade.         Atmospheres. 

Nitrogen           ...  —194°  ...  —146°  ...  35 

Oxygen             ...  —182°  ...  —112°  ...  50 

Ethylene           ...  -136°  ...  +9-2°  ...  58 

Carbon  dioxide  —80°  ...  31  -9°  ...  77 

Ammonia          ...  —39°  ...  130°  ...  115 

Chlorine             ...  —  34°  ...  141°  ...  84 

Sulphur  dioxide  — 10°  ...  155°  ...  79 

Ether     +35°  ...  195°  ...  40 

Alcohol             ...  78°  ...  235°  ...  67 

Water 100°  ...  365°  ...  200 

124.  Spheroidal  State. — If  a 'bright  metal  plate  be  heated 
to  a  sufficiently  high  temperature,  and  drops  of  water  be  allowed 
to  fall  on  it,  the  water  does  not  rapidly  evaporate,  but  forms 
globules  that  roll  and  spin  about.  The  water  passes  into  the 
spheroidal  state.  The  evaporation  is  very  slow,  and  the  state 
depends  upon  the  plate  being  retained  at  a  high  temperature. 
If  the  lamp  be  removed,  and  the  plate  cools  down,  a  point  is 
reached  when  the  globule  boils  rapidly  away.  The  metal,  for 
water,  should  be  above  150°  C.;  foralcohol,  130°;  and  ether,  60°. 

Measurements  have  shown  that  the  globule  is  always  below 
the  boiling  point  of  the  liquid.  Thus  water  was  found  to  be 
at  95°  C. ;  ether,  34°;  liquid  sulphur  dioxide,  — 11°. 

This  leads  to  a  remarkable  result.  If  a  platinum  vessel  be 
heated  to  redness,  and  liquid  sulphurous  acid  be  poured  into 
it,  the  acid  assumes  the  spheroidal  state,  and  therefore  its 
temperature  is  below  — 11°.  Drops  of  water  let  fall  into  the 
acid  are  at  once  frozen. 


Liquefaction  of  Gases— Vapour- Densities        185 

Faraday  mixed  solid  carbonic  acid  with  ether,  dropped 
it  into  a  red-hot  capsule,  when  it  assumed  the  spheroidal  state ; 
the  temperature  was  below  - 100°  C.  He  now  dropped  mercury 
into  the  spheroid,  when  it  at  once  solidified. 

The  spheroid  does  not  touch  the  metal — a  fact  readily 
shown  by  experimenting  on  a  level  plate.  A  lighted  candle 
can  be  seen  on  looking  between  the  plate  and  the  drop  (Fig.  72). 


FIG.  72. 

This  is  also  demonstrated  by  connecting  one  pole  of  a  battery 
(with  a  galvanometer  in  circuit)  with  the  plate,  and  dipping  the 
other  pole  into  the  spheroid.  The  absence  of  movement  of 
the  galvanometer  needle  shows  that  the  spheroid  and  metal 
do  not  touch  ;  it  rests  upon  a  cushion  of  its  own  vapour,  that 
is  continually  renewed,  and  which  transmits  heat  slowly,  and 
therefore  the  only  heat  that  reaches  it  does  so  by  radiation. 

The  fact  that  the  liquid  does  not  touch  the  solid  is- also 
shown  by  heating  a  ball  of  copper  to  redness,  and  lowering  it 
into  a  vase  of  water  at  about  99°  C.  An  envelope  of  vapour 
surrounds  the  metal  until  its  temperature  falls  to  140°,  when 
contact  takes  place,  and  the  water  boils  suddenly. 

If  the  hand  be  cooled  down  by  dipping  it  into  ether,  and 
then  be  dipped  into  hot  lead,  the  cushion  of  ether  vapour 
prevents  harm  taking  place,  provided  that  it  be  not  kept  long 
enough  for  the  vapour  to  evaporate.  By  damping  the  hand 
with  water,  it  can  be  dipped  with  safety  into  molten  shellac. 

125.  Vapour-Density. — The  process  of  finding  the 
vapour-density  of  a  substance  will  illustrate  many  parts  of  the 
preceding  chapter. 


1  86  Heat 

The  density  of  air  has  already  been  determined  (see  §51) 
as  0*001293  gram;  this  is  the  mass  of  i  c.c.  at  o°  C.  and  at 
760  mm.  pressure.  The  density  of  hydrogen  is  0-0000896. 
As  the  mass  of  i  c.c.  of  water  under  like  conditions  is  i  gram, 
the  above  numbers  express  the  relative  density  of  the  substance 
to  water. 

For  many  purposes  in  chemistry  and  physics  it  is  neces- 
sary to  know  the  density  of  the  vapour  of  a  liquid  or  solid. 
The  density  is,  as  a  rule,  compared  with  the  density  of  air  or 
hydrogen  under  like  conditions,  and  is  a  relative  density. 

The  principle  is  (a)  to  volatilize  a  known  mass  of  the  solid 
or  liquid,  and  measure  the  volume  of  the  vapour  produced 
under  the  conditions  of  temperature  and  pressure  ;  or  (b)  to 
measure  the  volume  of  a  certain  vapour,  and  then  determine  its' 
mass  under  the  observed  conditions  of  temperature  and  pressure. 

If  the  volume  of  air  be  vy  the  temperature  /,  and  the  pressure 
/,  then,  by  Boyle's  and  Charles's  laws,  the  mass  of  v  c.c.  =  w  = 

z>(o'ooi293)  ,        a/)  7  60'     I  lltre  of  dry  air  at  20°  C<  and  a 

.        1000(0-001293)  X4X76o 
pressure  of  4  atmospheres,  weighs  _ 


126.  Hoffmann's  Method.  —  Hoffmann's  is  a  modification 
of  Gay  Lussac's  method.  A  graduated  tube,  b  (Fig.  73),  called 
a  eudiometer,  about  40  inches  long,  is  filled  with  mercury  and 
inverted  in  a  trough  of  mercury,  g,  the  Torricellian  vacuum 
being  formed  above  the  mercury.  The  eudiometer  is  sur- 
rounded by  another  tube,  a.  The  liquid  or  solid  to  be  vaporized 
is  contained  in  a  very  small  stoppered  bottle,  h  (drawn  on 
a  larger  scale).  By  weighing  the  bottle  empty  and  again  con- 
taining the  liquid,  the  mass,  m,  of  the  substance  is  -known. 
The  stoppered  bottle,  placed  under  the  tube  in  g,  rises  to  / 
into  the  vacuum.  A  liquid,  whose  boiling  point  is  known, 
is  heated  in  e.  The  vapour  enters  the  second  tube  by  c, 
and  escapes  by  d  into  a  condenser.  The  increased  tem- 
perature with  diminished  pressure  is  sufficient  to  vaporize 
the  liquid.  The  advantage  of  the  Torricellian  vacuum  is  that, 


Liquefaction  of  Gases — Vapour- Densities        187 

being  under  reduced  pressure,  the  boiling  point  is  lowered 
(§  100),  and  thus  the  experiment  can  be  conducted  at  a 
much  lower  temperature  than 
would  be  possible  at  ordinary 
atmospheric  pressure. 

Water  boiled  in  e  gives  a  tem- 
perature in  a  nearly  at  100°  C., 
and  this  is  sufficient  to  vapo- 
rize substances  whose  boiling 
point  is  150°  under  ordinary 
atmospheric  pressure. 

The  following  readings  are 
taken : — 

(1)  The  temperature  of  the 
tube  (/);  that  is,   the  boiling 
point  of  the  liquid  in  e. 

(2)  The  height   (h)   of/ 
above  the  level  of  the  mercury 
in  g.     If  the  barometer  be  H, 
then   H  —  h  is  the  pressure 
to  which  the  gas  is  subjected. 

(3)  The  volume  (v)  of  va- 
pour formed  in   cubic  centi- 
metres, read  from  the  tube. 

Then  a  volume,  v,  of  a  sub- 
stance, at  a  temperature  t°  and 
pressure  H  —  h,  weighs ;;/  grams.  FIG.  73- 

i  c.c.  of  air  at  o°and  760  mm.  pressure  weighs  0*001293  gram; 
/.  i  c.c.  of  air  at  /°  and  (H  —  h)  mm.  pressure  weighs 

H  -  h         i 

0-001293  •-^-•rTT/ grams 

.'.  vapour-density  of  substance 

mass  of  vapour 
mass  of  equal  volume  of  air  under  like  conditions 

H  -h         i 
=  m  -4-  £'0*001293  • 


760 


at 


m(i  -f-  at)*] 60 

—  h) 


1 88  Heat 

Example.— Find  the  density  of  chloroform  vapour  (air  -  i) 
from  the  following  data  : — 

Quantity  of  chloroform  taken  =  0*096  gram 

Volume  occupied  in  tube  =  60  c.c. 

Temperature  of  vapour  =  30°  C. 

Barometric  pressure  reduced  to  o°  C.  =  752  mm. 
Height  of  mercury  column  in  tube     -  5067  mm. 

(i)  Substituting  in  formula,  we  obtain— 

IT  i      •  0-096(1  -f  0*003665  x  30)760  , 

Vapour-density  =  -  .  y  v       -. — °  "   .    =4*26 

60  x  0-001293(752  -  5067) 

Or  (2)  60  c.c.  of  dry  air  at  30°  C.  and  245-3  mm.  pressure 
weigh  6^_X  (0-001293)  X  245-3      0.02256 
760  X  (1-10995) 

/.vapour-density  = mass  of  vapour          =  _o^6     =  ^ 

mass  of  equal  volume  of  air     0-022  56 

Corrections. — (a)  The  height  of  the  mercury  column  in  the 
tube  should  be  reduced  to  o°  C. 

5067  mm.    of    mercury   at   30°  C.  =  ^°    J  — < — 

i  +  0-000181  X  30 

=  504  at  o°  C. 

.*.  with  this  correction,  vapour-density  =  4-21 

(b)  The  volume  60  c.c.  is  read  at  30°  C. ;  if  the  tube  is 
graduated  at  o°,  the  true  volume  is  60  -7- 1  4-  0-000025  x  30 
=  60  -f-  (i  4-  0-00075). 

.'.  4'2i  should  be  multiplied  by  i  -f-  0-00075  =  4'21  +  0-0031 
This  correction  only  affects  the  third  place  of  decimals, 
and  can  be  neglected. 

(c)  The  mercury  in  the  tube  depressed  by  the  pressure  of 
mercury  vapour  at  30°  ;  this  equals  o'o6  mm.,  again  a  quantity 
we  can  neglect.     If  the  tube  were  heated  to  a  temperature  of 
200°  C.,  the  pressure  would  be  19*9  mm.,  in  this  case  too  great 
to  be  neglected. 

Generally  z/4-  (i  +  kt)  is  the  real  volume  of  the  vapour; 
k  =  coefficient  of  expansion  of  glass.  If  the  temperature  be 
above  100°,  we  must  allow  for  the  pressure  of  mercury  vapour, 


Liquefaction  of  Gases  —  Vapour-  Densities        189 


/,  also  in  exact  experiments  the  mercury  in  the  tube  will  be 
practically  at  temperature  /°,  and  therefore  the  true  value  of 

//  will  be  —  j—  r.  =  ti  ;  b  =  coefficient  of  expansion  of  mercury. 
Therefore  a  more  exact  formula  would  be  — 


Vapour-density  = 


m 


i  -f 


0*001293 


760 
H -/*'- 


127.  Victor  Meyer's  Method,— A  glass  tube,  b  (Fig.  74), 
with  a  bent  connecting  tube,  a,  is  placed  in  a  bath,  <r,  which 
contains  a  substance  whose  boiling  point  is  constant,  and  is 
much  higher  than  the  boiling  point  of  the  substance  to  be 
vaporized.  The  top  is  closed  with  a  cork,  d.  The  liquid  in 
c  is  boiled  constantly  for  some  time;  this  is  known  when 
no  more  air  issues  from  the  end  of  a,  which  is  placed  under 
water  (or  mercury).  A  graduated  tube  is  filled  with  water 
and  inverted  over  a  in  g;  the  stopper 
d  is  removed,  and  a  small  glass  tube 
containing  a  weighed  amount  of  the 
substance,  whose  mass  is  m,  is  dropped 
in  (the  bottom  of  b  is  protected  with 
asbestos).  The  substance  passes  into 
vapour,  and  displaces  an  amount  of 
air  equal  in  volume  to  the  vapour 
formed ;  this  enters  the  graduated  tube. 
This  amount  is  a  fraction  of  the  volume 
of  the  tube  #,  and  being  generally 
heavier  than  air,  diffusion  does  not  act 
sufficiently  in  the  time  to  affect  the 
result. 

When  no  more  vapour  issues,  the 
graduated  tube  is  moved  with  the 
usual  precaution  to  a  deep  vessel  of 
water,  depressed  until  the  level  inside 
the  tube  is  the  same  as  that  of  the  water  outside ;  and  the 
volume  v  at  the  temperature  /°  of  the  room  and  the  height  of 
the  barometer  (H)  is  read.  The  pressure  of  water  vapour,  A', 
at  /°  must  be  considered,  and  is  read  from  tables. 


FIG.  74. 


Heat 
The  weight  of  a  volume,  #,  at  t°  and  pressure  H  —  //,  is  * 

TT     __       7/ 


Weight  of  equal  volume  of  air  =  v(o'o 

760(1  +  at) 


:.  vapour-density  =  —  •£- 
7       z/(o' 

If  the  vapour-density  is  to  be  calculated  for  hydrogen,  we 
must  take  the  weight  of  i  c.c.  of  hydrogen  at  o°  and  760  mm.  j 
this  is  0-00008936  gram.  The  expansion  for  glass  (the  graduated 
tube)  at  low  temperatures  can  be  neglected. 

The  following  is  an  actual  example  by  Meyer's  method, 
where  the  expansion  of  mercury  in  the  barometer  is  allowed  for  — 

(1)  Weight  of  substance,  0*0355  gram. 

(2)  Volume  of  vapour  in  tube,  5'!  c.c.  ;   temperature  of 
room  near  tube  when  measured,  14°  C.  ;  barometer,  755*6  mm. 
at  a  temperature  of  117°  C. 

(a)  755*6  mm.  mercury  at  117°  reduced  to  o°  is  754. 

(b)  Gas  in  tube  is  subjected  to  a  pressure  of  754  mm.  —  the 
pressure  of  aqueous  vapour  at  14°  j  this  is  12  mm. 

.*.  pressure  is  754  —  12  =  742 
Weight  of  5'i  c.c.  of  hydrogen  under  like  conditions 

5  'i  •  0*0000896  •  (754  —  12) 
760  •  (i  H-  0*0003665  •  14) 
.'.  vapour-density  of  substance 

_  °'°355  '  76°  •  (i  +  0-003665  x  14) 

5'i  •  742  •  0*00008936 
=  838    (H  =  i) 

In  practice  the  density  of  dry  air  has  been  determined,  and 
the  results  tabulated  for  various  temperatures  and  pressures  ; 

O*OOI2Q3(H  -\-  k) 

so  that  the  value  —  760(1  -f  at)      =  ^  can  be  obtamed  at 

m 

once,  and  the  density  is  — 
v/3 

Avogadro's  law,  that  equal  volumes  of  gases  under  similar 
conditions  as  to  temperature,  contain  an  equal  number  of 
molecules,  is  generally  accepted.  The  determination  of  the 


Liquefaction  of  Gases — Vapour- Densities        191 

vapour-density,  therefore,  gives  at  once  the  molecular  weight 
of  a  substance ;  e.g.  hydrogen  =  i,  the  following  results  are 
obtained : — 

Atomic  weights.  Molecular  weights. 

Hydrogen      ...         ...         i         ...         ...       2 

Oxygen  ...          ...       16  ...     32 

Water  9         18 

accepting  that  a  hydrogen  molecule  consists  of  2  atoms,  and 
that  we  represent  its  atomic  weight  by  i. 

Then  2  is  the  molecular  weight  of  hydrogen.  If  a  given 
volume  of  hydrogen  contains  2  units  of  mass,  then  an  equal 
volume  of  oxygen  under  similar  conditions  weighs  32  units, 
and  water  vapour  18  units.  We  cannot  determine  the  actual 
number  of  molecules  in  the  volume,  nor  is  it  necessary ;  for 
let  n  be  the  number — 

-  =  mass  of  i  molecule  of  hydrogen 

^=       „       i  „          oxygen 

18 

—  =       „        i  „          water  vapour 

.'.  the  mass  of — 

i  mol.  of  hydrogen  :  i  mol.  of  oxygen  :  i  moi.  of  water  vapour 
: :  2  :  32  :  18 

If  from  other  considerations  we  know  that  i  molecule  of 
oxygen  contains  2  atoms,  then  the  atomic  weight  of  oxygen 
will  be  1 6. 

EXAMPLES.    VIII. 

1.  What  is  meant  by  "saturated  "  and  "  unsaturated  "  vapour  ? 

2.  How  would  you   proceed  to   graph   the   isothermal  for  sulphur 
dioxide  at  the  temperature  of  the  room  ?     Compare  its  isothermal  with  that 
of  air. 

3.  Compare  the  molecular  condition  of  I  Ib.  of  water  at  100°  with 
i  Ib.  at  200°  C. 

4.  Describe  Andrews's  research  in  reference  to  the  critical  temperature 
of  carbonic   acid.      Explain   carefully   "critical   temperature,"    "critical 
pressure,"  and  "critical  volume." 


192  Heat 

5.  Explain  the  principle  of  Faraday's  method  of  liquefying  gases.     Why 
is  it  impossible  by  such  method  to  liquefy  oxygen  ? 

6.  Describe    Cailletet's    and   Pictet's    methods    of   liquefying    gases. 
Compare  the  principles  of  the  two  methods. 

7.  What  meanings  do  you  attach  to  "gas"  and  "vapour"? 

8.  To  change  water  at  100°  C.,  under  ordinary  conditions,  to  steam  at 
100°  takes  a  large  number  of  thermal  units,  and  time  is  necessary  to  give 
these  thermal  units  to  the  water ;  if  the  steam  passes  into  water,  there  is 
a  sudden  liberation  of  heat :   how  do  you  account  for  the  phenomenon, 
in  the  liquefaction  of  carbon  dioxide  by  Andrews's  or  Cailletet's  method, 
that  a  slight  change  in  temperature  or  pressure  produces  a  change  of  state, 
and  this  marked  liberation  of  heat  does  not  seem  -to  take  place  ? 

9.  How  can  solid  carbonic  acid  be  made  ?    Describe  its  appearance, 
and  its  use  in  producing  freezing  mixtures. 

10.  What  is  meant  by  the  spheroidal  state  of  water  ?    How  is  it  pro- 
duced ?    Explain  the  phenomenon. 

11.  Explain  "density."    How  have  the  densities  of  air  and  hydrogen 
been  determined  ? 

12.  Calculate  the  vapour-density  of  a  substance  from  the  following  data, 
obtained  by  Hoffmann's  method  (air  =  i) : — 

Weight  of  substance  =  0*25  gram 

Volume  of  vapour  =  35  c.c. 

Temperature  of  vapour  —  ioo°C. 

Corrected  barometer  in  room  =  750  mm. 
Height  of  mercury  in  tube      =  400  mm. 

(i)  Obtain  an  approximate  result,  and  state  exactly  what  corrections 
you  neglect.  (2)  Apply  corrections  necessary  to  give  a  correct  result  to 
two  places  of  decimals. 

13.  Calculate  the  vapour-density  of  a  substance,  from  the  following 
results  obtained  by  Meyer's  method  : — 

Weight  of  substance  —  o '  1 5  gram. 

Volume  of  air  collected  =  30  c.c. 

Corrected  height  of  barometer  =  756  mm. 
Temperature  of  room  =  20°  C. 


193 


CHAPTER     IX. 
HYGROMETRY. 

128.  Moisture  in  the  Air.— The  presence  of  water 
vapour  in  the  atmosphere  at  any  time  is  shown — 

(1)  By  exposing  chemical  substances  that  readily  absorb 
moisture,  such  as  calcium  chloride,  potassium  hydrate,  freely  to 
the  air.     These  salts  absorb  moisture,  and  ultimately  dissolve 
in  the  water  abstracted  from  the  air.      Strong  sulphuric  acid 
will  also  act  as  a  hygroscopic  substance.     Or — 

(2)  By  the  visible  condensation  of  the  vapour  on  bodies 
whose  temperature  is  sufficiently  lowered.     If  a  glass  carefully 
rubbed  dry  on  the  outside  be  filled  with  cold  spring  water 
outside  a  room,  moisture  is  generally  deposited  on  the  outside 
of  the  glass  when  it  is  brought  into  the  room.     Placing  ice  in 
such  a  glass  makes  the  experiment  more  decisive.     Even  in 
winter,  when  the   temperature  is   below  freezing  point,  the 
presence  of  moisture  can  be  shown  by  placing  in  the  glass  a 
mixture  of  ice  and  salt.     The  moisture  will  then  freeze  and  be 
deposited  as  hoar-frost.     In    experimenting  with  Wollaston's 
cryophorus  (§  in),  at  first  moisture  is  deposited;  the  moisture 
ultimately  freezes. 

The  expressions  in  every-day  use,  "  dry  air,"  "  moist  air,"  "  a 
dry  day,"  simply  express  the  effect  of  the  air  on  ourselves  and 
surrounding  objects.  They  give  no  information  as  to  the  exact 
amount  of  moisture  in  the  air;  for  example,  a  cubic  foot  of 
"dry  air"  in  the  tropics  contains  more  moisture  than  a 
similar  volume  of  air  in  England,  and  there  is  generally  more 
moisture  in  the  air  in  summer  than  in  winter, 

o 


194 


Heat 


If  the  temperature  of  the  air  be  sufficiently  lowered,  we  have 
seen  that,  at  a  particular  temperature,  condensation  of  aqueous 
vapour  takes  place  ;  the  air  at  this  temperature  is  saturated. 

The  temperature  at  which  the  air  is  saturated  with  aqueous 
vapour,  that  is,  the  temperature  at  which  condensation  begins, 
is  called  the  dew-point. 

When  the  temperature  of  the  air  is  greatly  above  its  dew- 
point,  we  call  the  air  dry;  this  occurs  both  in  summer  and 
winter.  The  dryness  generally  consequent  upon  a  cold  east 
wind  is  a  too  familiar  experience.  If  the  temperature  is  near 
the  dew-point,  the  air  is  "  damp." 

129.  Hygroscopes. — For  roughly  showing  the  state  of  the 
atmosphere,  hygroscopes  are  popularly  used.  Twisted  catgut, 
when  moistened,  untwists  and  thus  lengthens, 
shortening  again  when  it  dries.  This  explains 
the  action  of  toys  in  which  figures  move  out 
and  in  toy  houses,  according  as  the  state  of 
the  air  is  far  removed  from  or  near  the  dew- 
point,  and  denote  that  rain  is  improbable  or 
probable. 

In  Saussure's  hygrometer  (Fig.  75)  a  long 
human  hair,  <r,  cleaned  by  boiling  it  in  car- 
bonate of  soda,"  is  attached  to  a  frame  at  d, 
passes  over  a  pulley,  o,  and  is  kept  stretched 
by  a  small  weight,  p.  The  hair  contracts  or 
expands  according  as  the  air  becomes  more  or 
less  damp,  that  is,  according  as  the  tempera- 
ture is  near  to  or  far  from  the  dew-point.  Its 
movements  are  indicated  by  the  index.  At- 
tempts have  been  made  to  graduate  the  scale, 
o°  denoting  that  the  air  is  quite  free  from 
moisture,  100°  that  it  is  saturated.  These  attempts  have  not 
been  so  successful  as  to  warrant  its  use  as  a  scientific  instru- 
ment. (See,  however,  §  134.) 

Certain  chemical  salts,  notably  those  of  cobalt,  change  their 
colour  according  as  they  are  dry  or  damp ;  thus  a  toy  figure 
dressed  in  garments  dipped  in  such  a  salt  and  dried,  is  arrayed 
in  blue  on  dry  days  and  in  pink  on  damp  days. 


FIG.  75. 


Hygi'ometry  195 

130.  Relative  Humidity.  —  The  problem  in  hygrometry 
is  to  measure  the  actual  mass  of  aqueous  vapour  in  a  given 
volume  of  air  at  any  time  :  this  is  the  absolute  humidity  of  the 
air.  The  maximum  amount  of  vapour  that  the  same  volume 
of  air  could  possibly  contain  under  the  same  conditions  of 
temperature  and  pressure,  that  is,  the  amount  that  would  be 
present  if  the  air  were  saturated,  may  be  called  the  maximum 
humidity. 

The  ratio  between  the  absolute  and  maximum  humidity  is 
called  the  relative  humidity  of  the  air.  This  is  an  important 
ratio  in  meteorology. 

The  atmosphere  consists  of  the  mixture  of  air  and  aqueous 
vapour,  and  the  questions  in  hygrometry  are  simply  applications 
of  the  knowledge  possessed  of  the  laws  relative  to  the  mixture 
of  gases  and  vapours  (§  98). 

The  pressure  of  aqueous  vapour  at  any  temperature  is  the 
same,  whether  in  a  vacuum  or  in  a  space  already  filled  by  a 
gas.  If  air  and  aqueous  vapour  be  mixed,  the  total  pressure 
equals  the  pressure  due  to  the  air  plus  the  pressure  due  to  the 
vapour. 

Under  ordinary  atmospheric  conditions  the  pressure  is  never 
great,  and  it  has  been  shown  by  experiments  that  the  deviations 
from  Boyle's  law  are  so  slight  that  they  can  be  neglected. 
When  the  vapour  is  nearly  saturated  we  should  expect  the 
deviation  to  be  so  great  that  it  should  be  taken  into  account, 
but  the  experiments  of  Regnault  and  Herwig  have  shown  that, 
up  to  the  point  of  saturation  of  water  vapour  in  air,  Boyle's 
and  Charles's  laws  can  be  applied  without  any  marked 
inaccuracy. 

The  volume  is  generally  measured  in  cubic  metres,  so  that 
the  mass  of  unit  volume  at  o°  C.  and  760  mm.  pressure  =1293 
grams. 


/.  mass  of  i  c.m.  of  air  at  f  and/  pressure  =  -r~\ 

Since  air  and  the  vapour  within  the  limits  obey  the  gaseous 
laws,  the  relation  between  their  densities  will  be  constant.  It 
is  found  that  the  density  of  aqueous  vapour  is  0*622  times  that 


196  Heat 

of  air  under  similar  conditions  (f  is  a  convenient  approxi- 
mation). 

.'.  mass  of  i  c.m.  of  aqueous  vapour  =  I293  'P  '  °  622  grams 

(i  +  at)  •  760  6 

Returning  to  our  definition  of  relative  humidity,  we  find 
that  the  problem  is  to  calculate  the  relation  of  ;/z,  the  mass  of 
aqueous  vapour  actually  present  in  a  given  volume  (V,  in  cubic 
metres),  when  its  pressure  is  /  and  temperature  /°,  to  M,  the 
mass  of  vapour  the  same  volume  of  air  could  actually  contain 
if  it  was  saturated  with  aqueous  vapour  whose  pressure  would 
now  be  P. 


p  •  0*622 
m          (i  +  at)  •  760          p 
M  =  ¥1293.  P.  0-622  =  P 
(i  +at)-?6o 

That  is,  the  masses  are  as  the  pressures  of  the  vapour  under 
the  two  conditions. 

P  is  the  maximum  pressure  taken  from  the  tables,  the 
result  of  experiments,  p  is  difficult  to  determine  directly, 
but  is  readily  known  from  the  following  considerations  :  If 
we  imagine  a  volume  of  air,  V,  separated  from  the  rest  of 
the  atmosphere  by  an  invisible  film  ;  and  if  P  be  the  atmo- 
spheric pressure,  and  /  the  temperature;  then  by  Boyle's 

VP 
and  Charles's  laws,  for  this  mixture  of  air  and  vapour,      ,      , 

is  a  constant.     Seeing  that  during  an  experiment  P  does  not 
change  — 

V 


at 


is  a  constant 


Now  consider  the  aqueous  vapour  in  this  mixture.     The 
volume,  as  before,  is  V,  and  its  temperature  /.     Let  the  pressure 

be  /,  again    — 2—  =  constant  j  but      .     .  is  constant    by 
i  -p  at  T.  -\-  dt 

above, 

.*.  /  is  a  constant 


Hygrometry 


197 


It  is  therefore  only  necessary  to  determine  /  at  a  suitable 
temperature. 

The  method  followed  is  to  cool  down  the  air  until  condensa- 
tion just  begins,  and  note  the  temperature  of  the  dew-point. 
From  tables  the  pressure  at  this  particular  temperature,  and 
therefore  at  any  other  temperature,  is  known. 

Hygrometry  is  then  reduced  to  determining  the  dew-point. 

131.  Daniell's  Hygrometer.— The  bulbs  A  and  B  (Fig.  76), 
and  tube  connecting  them,  are  simply  Wollaston's  cryophorus, 
save  that  ether  is  substituted  for 
water,  and  that  a  delicate  ther- 
mometer is  inserted  in  the  instru- 
ment, so  that  its  bulb  comes  into 
the  middle  of  A ;  B  is  surrounded 
with  a  piece  of  muslin.  The  ether 
inside  is  all  run  into  A.  Ether 
is  then  dropped  on  B ;  it  rapidly 
evaporates;  the  temperature  of 
B  is  lowered,  and  the  ether  vapour 
inside  condenses ;  evaporation 
takes  place  from  A,  and  reduces 
the  temperature,  the  reduction 
being  recorded  by  the  thermo- 
meter. Soon  moisture  is  de-; 
posited  on  the  outside  of  A; 
that  is,  the  dew-point  (/)  is 
reached.  This  will  be  lower 
than  the  actual  dew-point  The 
dropping  of  ether  on  B  is  now 


FIG.  76. 


stopped,  and  the  temperature  of  A  rises  again.  When  the  dew 
has  disappeared,  the  temperature  /'  is  read.  The  mean  * 

is  called  the  dew-point.  To  render  the  deposition  more 
readily  visible,  A  is  made  of  black  glass.  It  is  an  assistance 
in  noting  the  point  to  draw  a  camel-hair  brush  across  A ;  the 
streaks  are  readily  seen.  The  temperature  of  the  air  is  given 
by  the  thermometer  on  the  stem. 

The  temperature  of  the  room  is  17-5°  C. ;  the  dew-point,  as 


1  98  Heat 

read  from  a  Daniell's  hygrometer,  is  the  mean  of  14°  C.  and 
14*5°  C.  :  find  the  hygrometric  state  of  the  air. 

Dew-point  =  H+-M1  =  I4.25°  C. 

f 
The  maximum  pressure  of  aqueous  vapour  at  17*5°    =  14-88 


,»        14*25°  =  12-01 


i  .•      i       -j-,.        m       P 

•••  relative  humidity  =  M  =  f  = 

The  instrument  is  not  very  reliable.  It  is  not  easy  to 
exactly  determine  when  the  moisture  appears  and  disappears. 
The  evaporation  of  A  is  at  the  surface,  and  the  thermometer 
dipping  into  the  liquid  may  not  give  the  exact  temperature. 
Glass  is  a  bad  conductor,  and  the  temperature  of  the  outside 
is  not  therefore  the  same  as  that  of  the  liquid.  The  observer 
also  modifies  by  his  presence  the  hygrometric  state  of  the  air. 
The  evaporation  of  the  ether  from  B,  seeing  it  contains  water 
vapour,  also  affects  the  result. 

132.  Dines's  Hygrometer.—  On  turning  a  tap,  B  (Figs. 
77  and  78),  water  flows  from  a  vessel,  C,  along  the  pipe  D. 


FIG.  77. 


A  diaphragm  in  the  chamber  R  compels  the  water  to  flow 
round  the  diaphragm,  pass  over  the  top,  and  escape  by  the 
end  pipe  A  (Figs.  78  and  79).  Above  the  diaphragm,  but  not 
touching  it,  is  the  bulb  of  a  delicate  thermometer,  T,  graduated 


Hygrometry 


199 


horizontally.  The  chamber  is  closed  above  by  a  piece  of  very 
thin  blackened  glass,  P,  which  is  rubbed  dry  before  any 
experiment.  The  water  in  C,  cooled  with  ice  if  necessary, 
and  kept  well  stirred,  is  allowed  to  run  by  turning  B.  As  soon 


SE  CTION 

FIG.  78. 


as  dew  is  deposited  on  the  blackened  glass,  the  thermometer 
is  read  (/°).  B  is  closed,  and  the  thermometer  read  again 
(t°)  when  the  film  disappears.  The  mean,  as  before,  is  the 
dew-point.  The  experiment  should  be  arranged  so  that  the 


THERMOMETER 


PIPE   D 


^PARTITION 
TO  DIRECT 
FLOW  OF     WATER 


ALL   FULL  OF 
FIG.  79. 


WATER 


temperature  of  the  water  in  A  is  not  much  below  the  dew- 
point.  By  care  the  temperatures  /°  and  t°  can  be  made  to 
almost  agree. 

133.  Regnault's  Hygrometer  —  The  glass  tubes  D,  E 
(Fig.  80)  are  closed  at  the  bottom  by  thin  plain  silver 
"thimbles."  A  thermometer  is  inserted  in  each,  T,  /.  D 
contains  ether  or  alcohol,  and  is  connected  with  a  reservoir, 
G,  by  the  indiarubber  tube  and  the  central  system  of  the 
apparatus.  G  is  filled  with  water.  On  allowing  water  to 
escape  from  G,  air  enters  at  A,  bubbles  through  and  evaporates 
the  liquid  in  D.  The  consequent  fall  in  temperature  causes  the 
deposition  of  dew,  and  the  temperature  of  T  is  read.  The 
tap  is  closed,  and  the  temperature  when  dew  disappears  is 


2OO 


Heat 


read.  These  two  are  brought  as  close  as  possible  together,  and 
the  mean  taken.  E  is  not  connected  with  D  or  G,  and  gives 
the  temperature  of  the  air  at  the  time. 


FIG.  80. 

The  hygrometer  is  readily  imitated  by  using  test-tubes 
(Fig.  81).  The  air  is  forced  through  by  a  pair  of  bellows. 
The  vapour  that  escapes  is  condensed  in  a  flask,  which  should 
be  further  removed  from  the  apparatus  than  is  shown  in  the 
figure,  and  kept  in  cold  water. 

134.  The  Wet  and  Dry  Bulb  Hygrometer. — In  the 
instruments  described,  a  distinct  experiment  has  to  be  per- 
formed each  time  the  dew-point  is  required ;  and,  as  we  have 
seen,  some  trouble  and  care  are  needed  to  make  a  good  deter- 
mination. The  advantage  of  the  wet  and  dry  bulb  thermometer, 
first  suggested  by  Leslie,  and  improved  into  its  present  form 
by  Mason,  is  that  the  readings  are  always  available  (Fig.  82). 
It  consists  of  two  delicate  thermometers  hanging  side  by  side. 
The  bulb  of  one  is  surrounded  with  well-cleaned  cotton,  which 


Hygrometry 


20 1 


also  dips  into  a  reservoir  of  water ;  this  bulb  is  therefore  always 
damp,  and  the  evaporation  from  its  surface  lowers  the  tempe- 
rature, so  that  its  temperature,  save  when  the  air  is  saturated, 
is  always  below  that  indicated  by  the  dry  bulb;  and  the 


FIG.  81. 


FIG.  82. 


further  removed  the  temperature  of  the  air  is  from  the  dew- 
point,  the  greater  will  be  the  evaporation  at  the  wet  bulb,  and 
consequently  the  lower  will  be  its  temperature. 
Apjohn's  formulae,  used  in  this  country,  are  — 


-       d        h 

F  =/  --  -  x  — 

96       30 

F  is  the  pressure  of  the  vapour  in  the  air  in  inches  ;  /  is 
the  maximum  pressure  of  aqueous  vapour  for  the  temperature 
of  the  wet  bulb  (taken  from  Regnault's  tables)  ;  d  is  the  differ- 
ence between  the  readings  of  the  wet  an<l  dry  bulbs  in  degrees 
Fahrenheit;  h,  the  barometric  height  in  inches.  The  value 


2O2  Heat 

87  is  used  above  32°  F.,  and  96  when  the  bulb  is  surrounded 
with  ice,  i.e.  below  32°  F. 

The  sudden  change  of  87  to  96  in  the  denominator  of  the 
fraction  at  the  freezing  point  suggests  the  uncertainty  of  the 
instrument  at  and  near  32°  F.  In  some  countries  the  hair 
hygrometer  (§  129),  carefully  graduated,  is  used  for  observations 
near  the  freezing  point. 

h  near  the  sea-level  will  not  differ  greatly  from  30 ;  therefore — 

.  .*-'-* 

gives  a  useful  approximation. 

Practically,  all  meteorological  results  are  recorded  in  England 
in  degrees  Fahrenheit  and  in  inches.  An  approximate  result, 
if  the  units  used  are  degrees  Centigrade  and  millimetres,  is — 

F  =/  —  0*00074^ 

Having  found  F,  the  relative  humidity  is  readily  calculated. 
If  we  require  the  dew-point,  it  is  only  necessary  to  find  the 
temperature  in  the  tables  opposite  to  the  found  value  of  F. 

The  results  obtained  from  these  formulae  have  been  checked 
over  a  long  series  of  years  with  actual  observations  made  with 
other  hygrometers. 

Glaisher  has  constructed  a  set  of  factors  for  finding  the 
dew-point  by  inspection.  For  their  use  we  have — 

M  4  -  k(td  -4) 

8  is  the  dew-point ;  td  is  the  temperature  of  the  dry  bulb ; 
tw  is  the  temperature  of  the  wet  bulb ;  k  is  a  factor  depend- 
ing upon  the  temperature  of  the  dry  bulb  at  the  time  of  the 
observation. 

GLAISHER'S  FACTORS. 

Dry  bulb  temperature.  Dry  bulb  temperature. 

F.  Factor.  F.  Factor. 

30-31  ...          4'I  45-5°°          —          2'1 

31-32°  ...  37  5°-55°  •••  2*0 

32-33°  •••  3'3  55-6o°  ...  1-9 

33-34°  •••  3'°  60-65°  ...  1-8 

34-35°  •••  2-8  65-70°  ...  r8 

35-40°  ...  2-5  70-80°  ...  17 

40-45°  ...  2-2  80-85°  •••  * '6 


Hygrometry 


203 


Sets  of  tables  based  upon  these  factors  and  corrected  from 
actual  observations  are  also  issued.  A  selection  from  these 
tables  follows : — 


Read- 

Difference between  dry  and  wet  thermometers. 

ing  of 

dry 

thermo- 
meter. 

1-0° 

2-0° 

3-0° 

4-0° 

5-o° 

6-0° 

7'0« 

8-0° 

9-0° 

IO'O° 

Amount  to  be  subtracted  from  the  wet  thermometer  to  obtain  the  dew-point. 

F. 
30° 

3-2 

6-3 

9'5 

12-6 

15-8      18-9 

22-1 

25-2 

28-4 

31-5 

31° 

2'7 

5  "4 

8*1 

10-8 

13-5       16-2 

i8"9 

21-6     24-3 

27-0 

32° 

2'3 

4-6 

7-0 

9'3 

1  1  -6      13-9 

16-2 

18-6     20-9 

23-2 

33° 

2'0 

4-0 

6-0 

8-0 

I0'0         I2'J 

14*1 

16-1      18-1 

20-1 

34° 

1-8 

3'5 

5'3 

7-1 

8-9    !  10-6 

12-4 

14*2     15-9 

177 

55° 

I'O 

1-9 

2-9 

3| 

4'8  1     5-8 

67 

77 

8-6 

9'6 

56° 

0-9 

1-9 

2-8 

47    |    5*6 

6-6 

7'5 

8;s 

9  '4 

57° 

0-9 

1-8 

2-8 

37 

4'6       5  "5 

7*4 

9-2 

58° 

0-9 

r8 

27 

4-5 

5'4 

6-4 

7-2 

8-2 

9-0 

59° 

0-9 

1-8 

27 

3-6 

4'5       5'3 

6-2 

7-1       8-1 

8-9 

60° 

0-9 

1-8 

2-6 

3'5 

4'4 

5'3 

6-2 

7-0       7-9 

8-8 

61° 

0-9 

1-7 

2-6 

3*5 

4*4  i      5'2 

6-1 

7-0       7-8 

87 

62° 

0-9 

17 

2-6 

3*4 

4'3        5'2 

6-0 

6-9       77 

8-6 

63° 

0-9 

17 

2-6 

3*4 

4*3 

5-1 

6-0 

6-8       77 

8*5 

64° 

0-8 

17     2-5 

3'3 

4-2 

5-0 

5-8  1    6-6       7-5 

8'3 

The  student  will  notice  that  the  three  methods  for  determin- 
ing the  dew-point  do  not  give  the  same  result.  The  theory 
cannot  yet  be  said  to  be  in  a  perfect  state,  and  close  agreement 
cannot  therefore  be  expected. 

135.  Example. — The  temperature  of  the  dry  bulb  is  58° 
F.,  that  of  the  wet  bulb,  55°  F. ;  the  barometer  stands  at  29 
inches  :  find  the  dew-point  and  the  relative  humidity. 

The  maximum  pressure  of  aqueous  vapour  at  55°  =  0*4329 
inch;  d  =  58  -  55  =  3°;  h  =  29. 

By  Apjohn's  formula — 

T,       ,.      d        h  *        20 


=  0-4329  -  0-0333  =  0-3996  inch 
From  the  tables  the  pressure  of  aqueous  vapour  at  52°  is 


2O4  Heat 

0-3882  ;  at  53°  it  is  0^4026  ;   therefore  the  temperature  of 
dew-point  is  52^1°  =  52-8°  C. 
By  Glaisher's  formula  we  have  — 

8  =  tt  -  k(td  -  4)  =  58-3^ 
The  factor  for  55°  is  1-9  ; 

/.  dew-point  =  58  -  57  =  52-3°  C. 
From  the  tables  we  obtain  at  once,  dew-point  =  55  —  27 

=  52'3. 

The  maximum  pressure  at  58°  =  0^4822  inch 

o"*oo6 
The  relative  humidity  =  Q.  %22  =  0-83 

136.  Chemical  Hygrometer.  —  The  chemical  method  is 
also  used  for  determining  the  relative  humidity. 

A  known  volume  of  air  is  drawn  through  drying-tubes 
containing  hygroscopic  substances,  such  as  calcium  chloride, 
pumice-stone  dipped  into  strong  sulphuric  acid.  The  increase 
in  the  weight  of  the  tubes  (;;/)  gives  the  amount  of  water 
vapour  in  a  known  volume  of  air  at  /°.  The  maximum  amount 
can  be  calculated  as  before,  M.  Then  — 

—  =  relative  humidit) 

137.  Weight  of  Given  Volume  of  Air  or  Gas.—  To 
find  the  weight  of  a  given  volume  of  atmospheric  air,  let  v  be 
the  volume  in  metres  ;  P,  the  barometric  pressure  ;  /,  the  tem- 
perature ;  and  flt  the  dew-point.     From  the  dew-point  we  can 
obtain  the  value  /  of  the  pressure  of  the  aqueous  vapour. 
By  the  law  of  the  mixture  of  gases  and  vapour,  we  have  — 

(a)  Weight  of  v  metres  of  dry  air  at  temperature  /  and 
pressure  P  —  /.  This 

V  X    I2Q3(P  — 

v 


(£)  Weight  of  v  metres  of  aqueous  vapour  at  temperature 
t  and  pressure  p  =  the  weight  of  an  equal  volume  of  dry  air 
multiplied  by  0*622. 


Hygrometry  205 

v  X  1293  x  P 

x  0-622  grams 


.-.  total  weight  =  a  +  b  =  760  <P  ~ 


For  example,  when  the  barometric  height  is  762  mm., 
the  temperature  of  the  room  20°  C.,  and  the  dew-point  16°  C., 
find  the  weight  of  10  cubic  metres  of  air. 

The  pressure  of  aqueous  vapour  at  16°  C.  =  13-5  mm. 

(a)  The  weight  of  10  cubic  metres  of  dry  air  at  20°  C.  and 
762  —  13-5  =  748*5  mm.  pressure 


=  11,865  grams 
The  weight  of  the  aqueous  vapour 


.*.  total  weight  =  11,978  grams  =  1  1*978  kilograms 


or  total  weight  =  10  fofr68  ~  °'*>&  * 

10  X  1293  X  273 

.93  x  760  "  *  "6'897 
=  1  1  -9  7  8  kilograms 

Remembering  that  hygrometric  state  =  e  =  ^,  if  we  know 
e,  then  for  /  we  can  substitute  eP. 

WORKED  EXAMPLE. 

1250  c.cm.  of  hydrogen  are  collected  over  water,  the  temperature  of  the 
water  is  18°  C.,  and  the  height  of  the  water  inside  the  collecting-jar  is 
10  cm.  above  the  level  of  the  water  in  the  pneumatic  trough  :  find  the 
mass  of  dry  hydrogen  collected.  The  corrected  barometer  reads  756  mm. 

(a)  The  maximum  pressure  of  aqueous  vapour  at  18°  C.  =  15  36  mm. 

(b)  The  density  of  water  at  20°  C.  =  0*9983 

,,  mercury  at  o°  =13-596 


206  Heat 

.'.  a  column  of  water  20  cm.  high  is  equivalent  to  a  column  of  mercury 

at  o°  C.  =  —  rgQ6 —  cm*  =  r47  cm*  =  I4'7  mm§  *"gh 
(f)  The  total  pressure  inside  the  collecting-jar  =  756  —  147  =  741-3  mm. 
.*.  the  dry  gas  is  subject  to  a  pressure  of  741*3  —  15*36  mm.  =  725*94  mm. 
(d)  1250  c.cm.  of  hydrogen  at  20°  C.  and  725*94  mm.  pressure  weighs 
1*250  x  0-0896  X  273  x  725'94 

293  x  760  -  =  0-0997  gram. 

EXAMPLES.    IX. 

1.  What  is  meant  by  the  dew-point  ?    How  does  the  dew-point  show 
the  amount  of  vapour  in  the  atmosphere  ? 

2.  Describe  the  action  of  Daniell's  hygrometer.     The  temperature  of 
the  air  is  18°  C.,  the  dew-point  is  12°  C. :  determine  the  hygrometric  state 
of  the  atmosphere. 

3.  What  is  meant  by  the  term  "humidity,"  or  "hygrometric  state  of 
the  atmosphere  "  ?   The  dew-point  on  a  certain  day  is  1 2°  C.,  the  temperature 
of  the  air  being  16*5°  C.  :  find  its  humidity. 

4.  Find  the  mass  of  5  litres  of  air  when  the  temperature  is  15°,  the 
dew-point  1 1  '6°,  and  the  barometric  height  is  75  '8  mm, 

5.  What  are  the  advantages  of  the  wet  and  dry  bulb  hygrometer 
compared  with  a  Daniell's  hygrometer  ? 

6.  The  air  in  a  laboratory  is  16°  C. ;  5  c.m.  of  air  are  drawn  through  a 
chemical  hygrometer,  and  the  drying-tubes  show  an  increase  of  50*4  grams  : 
find  the  hygrometric  state  of  the  room,  and  the  dew-point. 

7.  In  an  experiment  150  c.cm.  of  air  are  collected  over  mercury— the 
quantity  is  read  after  forcing  up  the  tube  sufficient  water  to  form  a  thin 
layer — the  barometer  in  the  room  reads  760  mm. ;  the  thermometer,  25°  C. ; 
the  mercury  in  the  tube  is  300  mm.  above  the  mercury  in  the  trough  : 
(i)  find  the  mass  of  dry  air;    (2)  determine  its  volume  at  o°   C.    and 
760  mm.  pressure. 

8.  On  Monday  the  dry  bulb  reads  60°  F.,  the  wet  bulb  52°  F.,  the 
barometer  7*58  mm.     On  Tuesday  the  dry  bulb  reads  58°  F.,  the  wet  bulb 
53°  F.,  the  barometer  761  mm.     Using  Apjohn's  formulae,  compare  the 
hygrometric  states  on  the  two  days. 

9.  Describe  Dines's  hygrometer.      Compare  it  with   Regnault's  and 
with  the  wet  and  dry  bulb  hygrometer. 


207 


CHAPTER  X. 
TRANSMISSION  OF  HEAT. 

138.  Convection. — In  the  experiments  on  ebullition 
(§  99)>  particles  of  the  liquid  were  heated,  and,  under  the  action 
of  gravity,  rose,  heavier  particles  taking  their  place.  Heat  was 
in  this  experiment  transferred  from  one  part  of  the  substance 
to  another  by  convection.  The  convection  currents  are  readily 
observable  if  a  few  pieces  of  bran  or  other  light  substance  be 
thrown  into  the  water. 

Transference  of  heat  by  convection  is  the  method  followed 
in  heating  dwellings  with  hot  water;  it  is  seen  on  a  large 
scale  in  the  transference  of  heat  by  the  Gulf  Stream.  Similar 
currents  are  observable  in  gases ; 
the  ascent  of  the  heated  par- 
ticles of  air  above  a  lamp,  the 
rise  of  heated  air  causing  air- 
currents,  are  examples. 

Convection  currents  may  as 
readily  be  induced  in  fluids  by 
cooling  one  part.  In  Fig.  83, 
if  ice  be  placed  in  the  test-tube,  the  water  near  it  becomes 
denser,  and  sinks,  and  currents  ensue.  This  illustrates  the 
effect  of  the  polar  cold  in  causing  ocean  currents,  their  direc- 
tion of  flow  being  modified  by  the  rotation  of  the  earth  and  the 
shape  of  the  land. 

Convection  of  air  is  illustrated  by  ventilation  and  the 
draught  of  chimneys.  The  heating  of  the  air  reduces  the 
density,  and  currents  are  induced. 


208  Heat 

139.  Conduction  of  Heat.— When  solids  are  heated,  the 
heat  is  transferred  from  one  point  to  another  by  conduction. 
The  end  of  the  poker  placed  in  the  fire  rises  in  temperature, 
the  near  particles  are  heated,  and  so  heat  is  ultimately  trans- 
ferred to  the  end.  Metals  generally  are  good  conductors  of 
heat,  while  felt,  wool,  etc.,  are  examples  of  bad  conductors. 

We  may  roughly  compare  the  conducting  power  of  two 
solids,  say  iron  and  copper,  by  placing  two  similar  rods  end  to 
end,  and  heating  the  junction.  After  some  time  we  can  try 
the  furthest  point  from  the  junction  where  a  match  will 
ignite  without  friction.  In  an  experiment  the  distance  was 
12  inches  on  the  copper  rod  and  6  inches  on  the  iron.  Or 

we  may  find  the  point  where 
paraffin  will  just  melt.  Cop- 
per, we  conclude,  is  a  better 
conductor  than  iron. 

A  cylinder  is  made  up 
(Fig.  84)  of  two  cylinders,  of 
wood  and  brass.  Paper  is 
tightly  wrapped  round  it,  and 
a  flame  is  applied  at  the  junc- 
tion of  wood  and  brass.  The 
FlG<  84-  brass  conducts  away  the  heat 

so  rapidly  that  the  paper  is  not  scorched.     Wood  is  a  bad 
conductor,  and  the  paper  round  it  is  at  once  burnt. 

Liquids  absorb  heat  readily ;  convection  currents  are  also 
formed,  so  that  fresh  masses  are  brought  under  the  influence 
of  the  heat.  For  this  reason  water  can  be  readily  boiled  in 
egg-shells  or  cones  made  of  paper;  lead  may  be  melted  in 
a  pill-box, — if  a  piece  of  paper,  however,  be  dipped  into  the 
molten  lead,  it  is  at  once  charred. 

The  heat  is  conducted  through  the  envelope  to  the  water 
or  lead ;  it  is  there  used  to  raise  the  temperature ;  the  water 
boils,  as  the  temperature  cannot  rise  above  100°  C. ;  the  heat 
is  then  utilized  in  evaporating  the  water.  A  slightly  altered 
explanation  applies  to  the  lead. 

Solids  may  be  readily  arranged  in  order  of  their  power  of 
conducting  heat,  by  the  following  experiment : — 


Transmission  of  Heat  209 

A  number  of  similar  small  cylinders  of  same  height  and 
diameter  are  made  of  copper,  brass,  bismuth,  wood,  cork,  etc. 
The  simple  air-thermometer  is  clamped  (Fig.  85).  One  cylinder 
of  copper  is  placed  in  boiling  water  for  some  time,  and  is 
used  always  in  the  experiment  as  the  heater.  Begin 
with  the  cylinder  of  lead,  and  place  it  on  the  top  of  the 
air-thermometer,  then  place  above  it  the  heater  at 
100°  C. ;  the  heat  is  conducted  through  the  lead,  heats 
the  air  in  the  thermometer,  and  the  liquid  sinks.  Wait 
until  it  reaches  its  lowest  point,  and  note  the  position. 
Remove  the  copper  to  its  bath,  take  away  the  lead,  and 
wait  until  the  liquid  rises  to  its  original  position.  Place 
now  a  second  cylinder,  say  bismuth,  on  the  top,  and 
repeat  the  experiment.  The  greater  the  amount  of  heat 
conducted,  the  greater  will  be  the  expansion  of  the  air, 
and  the  further  will  the  liquid  in  the  tube  be  depressed. 

The  following  will  be  the  order,  beginning  with  the 
best  conductor  :  (i)  copper ;  (2)  brass ;  (3)  iron ;  (4) 
lead;  (5)  bismuth;  (6)  wood;  (7)  cork. 

A  little  consideration  will  show  that  this  rough  exper  merit 
admits  of  many  inaccuracies.  The  heat  from  the  heater  has 
first  of  all  to  raise  the  temperature  of  the  cylinder  experimented 
upon ;  this  depends,  not  upon  its  power  of  conducting  heat, 
but  upon  its  capacity  for  heat.  It  is  only  after  the  substance 
ceases  to  rise  in  temperature  that  heat  will  be  transmitted 
through  it  in  a  way  that  we  can  say  is  due  to  conduction. 
We  have,  in  fact,  to  discriminate  between  rise  of  temperature 
and  conduction  of  heat. 

In  all  such  experiments  relative  to  conductivity  we  must 
see  that  the  temperature  of  the  substance  is  steady  before  we 
measure  conductivity ;  that  is,  the  flow  of  heat  must  be  steady. 
The  student  will  now  see  the  force  of  the  phrase,  after  some 
time,  used  in  this  section. 

140.  Ingenhaus's  Apparatus  is  a  metal  trough  with 
rods  of  various  materials,  but  of  the  same  diameter  and  length 
passing  through  corks  in  the  side  (Fig.  86).  The  rods  project 
the  same  distance  into  the  trough,  and  are  coated  outside 
with  wax.  Hot  water  is  poured  into  the  trough,  and  the 

p 


21(3 


Heat 


rate  at  which  the  melting  of  the  wax  travels  along  each  rod 
is  observed ;  this  is  evidently  observing  the  rise  of  temperature. 
After  the  flow  of  heat  is  steady,  the 
distance  melted  along  each  rod  is 
measured.  The  greater  the  distance, 
the  greater  the  conducting  power  of 
the  solid. 

141.  Despretz's  Method.— 
Despretz  formed  the  solids  on  which 
he  experimented  into  similar  long 
bars  (Fig.  87),  A,  B;  one  end  was 
turned  and  dipped  into  a  bath  of  molten  metal  that  gave 
a  constant  source  of  heat.  Thermometers,  /,/',...  /",  were 
inserted  into  small  cavities  in  the  bar,  at  equal  distances 
apart ;  the  cavities  were  filled  up  with  mercury.  A  screen,  S, 
protected  the  bar  from  the  heat  of  the  flame.  Readings  were 
taken  when  all  the  thermometers  remained  stationary.  If  the 


FIG.  86. 


FIG.  87. 

thermometers  be  of  similar  construction;  and  the  tops  of 
the  columns  of  mercury  be  joined,  a  curve,  a,  #',...  a",  is 
formed. 

The  heat  passing  through  any  section,  say  a  section  near 
the  thermometer  /",  is  now  a  constant  quantity,  and  f  ceases 


Transmission  of  Heat  211 

to  rise,  because  the  heat  passing  through  the  section  is  just 
sufficient  to  balance  the  heat  lost  in  the  bar  from  /'  to  B  by 
radiation  and  by  convection  (air-currents).  By  subtracting 
the  general  temperature  of  the  room  from  the  temperature 
indicated  by  each  thermometer,  we  obtain  the  excess  of 
temperature  of  each  point  above  the  rest  of  the  room.  Let 
us  suppose  the  rod  is  divided  into  eight  equal  parts,  then  if 
we  erect  perpendiculars  at  each  point,  proportionate  to  the 
excess  of  temperature,  we  can  plot  out  a  curve  that  graphi- 
cally shows  the  results  o  f  each  experiment. 

Despretz  gave  as  the  result  of  his  experiment  the  law 
that,  the  distances  along  the  rod  being  in  arithmetical  pro- 
gression, the  ordinates,  measuring  the  excess  of  temperature, 
decrease  in  geometrical  progression. 

For  example,  if  the  distances  of  /,  /',  etc.,  from  A  be  i,  2, 
3, ...  7  feet  respectively,  then  the  excesses  of  temperature 
above  the  room  for  each  thermometer  will  be  something  like 
the  following : — 

Distance      i        2       3       4       5       6)     7 
Excess        100     70     49     34     23     16     n 

where  g-  =  ^   =  g  etc.,  =  ^approximately. 

This  law  does  not  admit  of  general  application ;  it  must  be 
restricted  to  good  conductors. 

By  such  observations  the  relative  conductivities  of  metals 
were  determined. 

142.  Wiedemann  and  Franz's  Method. — We  have 
seen  that  the  flow  of  heat  must  be  steady ;  but  as  the  experi- 
ments depend  upon  the  radiation,  the  surface  of  the  bars 
must  be  similar  and  equal  in  each  experiment.  This  latter 
condition  was  attained  by  Wiedemann  and  Franz  by  making 
bars  exactly  equal  in  dimensions,  and  by  electroplating  the 
surfaces;  they  further  improved  the  method  by  using  small 
thin  bars  that  admitted  of  greater  care  being  taken  with  regard 
to  their  purity  and  condition  as  to  annealing,  etc.  Instead  of 
the  high  temperature  (that  of  molten  metal)  at  one  end,  as 
in  the  case  of  the  previous  experiments,  it  was  found  sufficient 


212  Heat 

to  keep  one  end  at  100°  C.  A  small  specially  constructed 
thermo-pile  (§  152)  was  used  to  determine  the  temperature  of 
the  various  points  of  the  bars,  thus  avoiding  the  break  in  the 
continuity  of  the  bars  caused  by  the  holes  for  the  thermo- 
meters. The  small  bars  were  further  placed  in  strong  cylindrical 
glass  vessels  which  were  surrounded  by  water;  the  bars  were 
therefore  experimented  upon  in  an  enclosure  whose  conditions 
could  be  kept  constant.  The  air  in  the  glass  vessel  could 
also  be  exhausted,  so  that  observations  could  be  made  in 
vacuo,  eliminating  thereby  the  effect  of  air-currents. 

These  two  experimenters  constructed  a  table  of  relative 
conductivities,  taking  the  conductivity  of  silver,  the  best  con- 
ducting metal,  as  100. 

TABLE. 

Silver  ...  ioo-o  Steel    11*6 

Copper  ...  73'6  Lead 8*5 

Gold  ,..  5  3  '2  Platinum      ^:.V  8-4 

Brass  ,.,,  23*1  Rose's  alloy     ...  2-8 

Tin  „.>'-  T4'5  Bismuth           ..."  1-8 

Iron  .:.  '.\  1 1 -9 

The  experimental  work  in  connection  with  conductivity  is 
difficult ;  slight  impurities  and  differences  in  physical  condition 
greatly  affect  the  results.  The  following  table,  due  to  Gracet- 
Calvert  and  Johnson,  will  show  this  clearly  : — 

Relative  conducting  power. 
Silver  =  1000. 

Silver     ...  1000 

Copper,  rolled ...         .-.;'                                ...  845 

Copper,  cast     ...        •....-      811 

„          ,,    with  o *5  per  cent,  of  arsenic    ...  669 

»  55  55          *  55  55  •"  57° 

Antimony,  cast  horizontally   ...         ...         ...       215 

„  „    vertically        192 

143.  Conductivity. — Such  numbers  are  useful  for  com- 
parison only.  In  order  to  give  definiteness  to  the  idea  of 
conductivity,  the  following  conditions  are  imagined :  The 


Transmission  of  Heat  215 

substance  whose  conductivity  we  wish  to  measure  is  supposed 
to  be  made  into  a  plate,  i  centimetre  thick  and  of  unlimited 
area.  The  two  sides  are  each  kept  at  a  constant  temperature, 
one  being  i°  above  the  other.  When  the  flow  of  heat  is  steady, 
the  quantity  of  heat  that  passes  through  one  square  centi- 
metre far  removed  from  the  edges  of  the  plate  of  the  sub- 
stance in  one  second,  measures  the  thermal  conductivity  of  the 
substance. 

The  quantity  of  heat  (Q)  transmitted  will  vary  (i)  as  the 
area  of  the  surface  (A)  ;  (2)  as  the  time  (/)  ;  (3)  as  the  differ- 
ence in  the  temperatures  of  the  two  faces  (6)  •  and  (4)  it  will 
vary  inversely  as  the  thickness  (d}.  That  is  — 

_       A/0  A/0 


where  k  is  a  factor  whose  value  will  change  for  each  substance. 
With  unit  area  (i  sq.  cm.),  unit  time  (i  sec.),  unit  thick- 
ness (i  cm.),  and  unit  difference  in  temperature  — 

Q  =  * 

Exampk.  —  How  much  water  will  be  evaporated  per  hour 
when  it  is  boiled  at  ioo°in  an  iron  boiler  1-5  cm.  thick,  having 
the  area  of  the  heating  surface  equal  to  460  sq.  cm.,  and  its 
outer  surface  kept  at  180°  C,  the  conductivity  of  iron  being 
0*175  (Day's  Examples)?  (See  §,  149.) 

7  A/0       0-175  X  460  X  3600  x  80 
Quantity  of  heat  =  Q  =  fc-j-  =  -  —,  —  - 

Q      0-175x460X360x80 
/.  mass  evaporated  =—  =  1*5x537  =28782grms. 

Seeing  that  in  such  examples  as  the  above  all  the  quan- 
tities save  k  can  be  determined,  it  seems  that  some  modifica- 
tion might  be  adopted  for  measuring  k  ;  and  this  method  has 
been  followed  by  Peclet  and  others.  It  is  found,  however, 
in  practice  that  such  methods  are  unsuitable  for  exact  deter- 
minations. 

The  change  in  the  values  of  the  coefficients  of  expansion, 
in  the  specific  heat,  etc.,  as  the  temperature  changes,  will 
suggest  also  that  conductivity  will  vary  with  the  temperature. 


214  Heat 

144.  Forbes's  Method. — The  principle  of  the  method 
used  by  Forbes  is  as  follows  : — 

One  end  of  a  bar  is  kept  at  a  permanent  temperature,  as 
in  Fig.  87.  Small  holes  are  drilled  at  various  distances  along 
the  bar,  in  which  thermometers  are  inserted,  the  holes  being  then 
completely  filled  by  a  few  drops  of  mercury.  After  several 
hours  the  temperature  of  each  part  of  the  bar,  as  indicated  by 
the  thermometers,  is  steady.  The  quantity  of  heat  that  passes 
any  section  (C)  of  the  bar  (Fig.  88)— whose  temperature  is 


UL 


FIG.  88. 

known — is  equal  to  the  heat  lost  by  radiation  and  the  con- 
vection of  air-currents  by  the  part  C  L.  This  quantity  is 
known  if  we  can  determine  the  quantity  of  heat  lost  by  each 
of  the  lengths  Ca,  at,  be,  .  .  .  pq>  etc. ;  we  have  simply  to  add 
these  quantities  together. 

In  order  to  determine  these  quantities,  a  preliminary 
experiment  is  made  by  heating  an  exactly  similar,  but  shorter, 
bar  to  a  high  temperature,  inserting  a  thermometer,  and  allow- 
ing the  bar  to  cool  under  similar  conditions  to  those  ex- 
perienced by  the  experimental  bar.  The  temperature  of  the 
bar  is  noted  at  certain  intervals,  frequently  at  first,  and  less 
frequently  as  the  bar  cools. 

By  this  experiment  we  know  the  number  of  seconds,  »,  it 
takes  the  bar  to  cool  from  t°  to  /!°. 

.'. =  r  -  rate  of  fall  of  temperature 

And  by  taking  /x  as  close  as  possible  to  /,  we  can  determine 
the  rate  of  fall  of  temperature  at  /°  (see  paragraph  illustrating 
Figs.  89  and  90). 

If  s  be  the  specific  heat  of  the  metal  at  that  temperature, 
and  m  the  mass  of  unit  length — 

Then  smr  -  loss  of  heat  per  second  per  unit  length  at  f 
From  the  data  a  table  is  constructed,  and  from  it  we  can 


Transmission  of  Heat 


215 


at  once  determine  the  quantity  of  heat  unit  length  of  the 
bar  will  lose  in  unit  time  at  any  temperature  within  the  limits 
of  the  experiment." 

Then  in  the  experiment  we  know  the  temperature  of/  and 
q  (Fig.  88),  and  its  length,  and  from  the  results  obtained  from 
the  cooling  of  the  short  bar  we  can  determine  the  amount  of 


771. 


FIG.  89. 

heat  radiated  By  calculating  for  every  such  distance,  taking 
the  distance  as  small  as  possible,  and  adding,  we  obtain  the 
amount  of  heat  that  has  passed  the  section  C. 

If  MN  (Fig.  89)  represents  the  excess  of  temperature  of  the 


FIG    90, 


2i6  Heat 

thermometer  at  M  above  the  temperature  of  the  room,  and  mn 
the  excess  of  the  thermometer  at  m,  then  the  temperature  has 
fallen  MN  —  mn  =  /N  for  a  distance  Mw,  and  therefore  the 

/N 
fall  per  unit  of  length  is          j   that  is,  the  tangent   of  the 


angle  N#/  measures  the  rate  at  which  the  temperature  is 
decreasing.  If  m  be  very  near  to  M,  the  line  joining  N« 
becomes  AC,  the  tangent  to  the  curve  at  N  (Fig.  90).  This 
tangent  makes  an  angle  <j>  with  the  bar. 


Q  =  k  (d'ffere"^°f  temperature^,  =  ^  ^ 
V  thickness  / 


If  we  take  A  as  unit  of  area,  and  /  unit  of  time  — 
Q  =  k  tan  <£ 

(1)  The  curve  can  be  constructed  from  experiments  (see 
beginning  of  §  141)  ; 

.*.  $  is  known  for  any  temperature 

(2)  We  have  shown  that  Q  can  be  determined  by  adding 
the  amounts  of  heat  radiated  by  each  small  portion  of  the  bar  ; 

.*.  k  can  be  determined  for  any  given  temperature 

145.  Thermometric  Diffusivity.  —  In  the  definition  given 
in  §  143,  the  flow  of  heat  is  steady;  that  is,  no  part  is  used  for 
raising  the  temperature  of  the  bar.  In  making  observations 
upon  conductivity  it  is  not  always  possible  to  have  a  steady 
condition  of  temperature  in  every  part  of  the  bar,  and  results 
have  to  be  obtained  from  readings  made  when  the  temperature 
is  rising  or  falling,  as  for  example  in  the  fall  of  the  temperature 
of  the  small  bar  in  Forbes's  experiment.  The  change  of 
temperature  not  only  depends  upon  the  c6nductivity,  but  also 
upon  the  capacity  of  the  body  for  heat.  Part  of  the  heat  only 
is  transmitted  from  particle  to  particle  ;  the  remaining  part  is 
used  in  raising  the  temperature  of  the  various  parts  of  the  bar. 
The  greater  the  capacity  of  a  substance  for  heat,  the  less  will 
be  the  rise  of  temperature  for  a  given  supply  of  heat;  the 
greater  the  conductivity,  the  greater  will  be  the  rise  in  tem- 
perature ;  that  is,  we  measure  a  quantity  (K)  that  varies  directly 


Transmission  of  Heat  217 

as  the  conductivity,  and  inversely  as  the  thermal  capacity,  so 

k 
that  K  =  -.    f,  the  thermal  capacity  of  unit  volume,  equals  the 

product  of  the  density  and  the  specific  heat 

When  the  flow  of  heat  is  not  steady,  what  is  measured  is 

k 
the  ratio  of  k  to  the  thermal  capacity ;  that  is,  -.     This  ratio 

is  called  "the  coefficient  of  thermometric  conductivity"  by 
Maxwell;  and  "the  coefficient  of  thermal  diffusivity"  by 
Thomson. 

In  determining  K,  we  measure  the  effect  that  a  quantity  of 
heat  in  passing  through  unit  volume  of  the  substance  pro- 
duces on  the  temperature  of  this  volume.  It  was  this  that  lead 

k 
Maxwell  to  give  to  K  =  -  the  name  "thermometric  conductivity," 

giving  the  name  "  calorimetric  conductivity  "  to  k.  Thomson 
gave  the  name  "  thermal  diffusivity  "  from  the  analogy  between 
the  coefficient  as  defined  and  the  coefficient  that  expresses 
the  rate  at  which  diffusion  takes  place  between  two  liquids. 

The  student  should  carefully  observe  which  coefficient  is 
given  in  examples. 

Forbes,  by  his  experiments,  found  the  conductivity  of 
iron  at  various  temperatures.  He  experimented  upon  two 
bars,  one  ij  inch  square,  the  other  i  inch  square.  His  units 
were  foot,  minute,  and  the  heat  required  to  raise  one  cubic 
foot  of  iron  one  degree  C. 

Thermometric  conductivity  or  diffusivity  — 
Temperature.  ii-inch  bar.  i-inch  bar. 

o°  C.  ...  0*01337  •••  0*00992 

50°  C.  ...  0*01144  ...  0*00904 

100°  C.  ...  001012  ...  0*00835 

200°  C.  ...  0*00876  ...  0*00764 

275°  C.  ...  0*00801  ...  0*00724 

Forbes  noticed  that  the  order  of  the  metals,  with  respect 
to  their  thermal  diffusivity,  was  the  same  as  their  order  with 
respect  to  their  electrical  conductivity;  his  experiments  on 
the  iron  bars  showed  that  the  thermal  diffusivity  decreased  as 


2i8  Heat 

the  temperature  rose,  analogous  to  the  decrease  in  electrical 
conductivity  following  a  rise  in  temperature.  This  suggested 
that  the  relation  would  be  the  case  with  all  metals. 

k 
Having  determined   the    thermal   diffusivity,  K  =  -,  the 

absolute  conductivity  can  be  calculated  by  multiplying  k  by  c. 
For  example,  the  thermal  diffusivity  of  a  specimen  of  copper 
at  20°  is  1*164. 

c  =  specific  heat  X  density  =  0*0933  x  8*9  =  0-830 

This  would  give  the  absolute  conductivity  as  1-164  x  0*830 
=  0*966. 

The  units  are  the  centimetre,  gram,  second. 

Absolute  conductivity.  Absolute  conductivity. 

k  k 

Copper  ...     1*108  Iron     0-164 

Zinc     0*307  German  silver...     0*109 

Brass   ...         ...     0-302  Ice      0-0057 

The  tables  give  results  without  specifying  the  temperature. 
The  temperature,  however,  as  we  have  seen,  slightly  affects 
the  results,  especially  in  metals,  and  the  result  should  probably 
be  of  the  form  a(i  4-  bt). 

There  is  considerable  disparity  in  the  results  of  different 
experimenters.  This  is  markedly  the  case  in  the  effect  of 
temperature.  Forbes  gave  the  general  result  that  the  con- 
ductivity of  all  metals  decreased  with  rise  of  temperature. 
Tait,  experimenting  with  the  same  bar,  found  that  all  save  iron 
increased  in  conductivity  as  the  temperature  rose;  Mitchell, 
again,  using  the  same  bar,  that  the  conductivity  of  iron,  like 
other  metals,  increased  with  rise  of  temperature,  and  this 
seems  to  be  the  true  law. 

146.  Dimensions  of  Conductivity  .—The  quantity  of 

heat  Q  =  k  —^  (§  143),  substituting  the  dimensions  of  area, 
time,  temperature,  and  thickness 


Transmission  of  Heat  219 

The  dimensions  of  conductivity  will  depend  upon  the  units 
used  in  measuring  heat  [Q]. 

(a)  Heat  measured  in    Thermal   Units. — The   dimensions 
=  MA  (§  75).     Conductivity  is  called  thermal  conductivity, 
or  absolute   conductivity,   and  is  represented  by  £,  whose 
dimensions  are — 

MA-M,TA=~ 

(b)  Heat  measured  in  Thermometric  Units  ;  that  is,  the  heat 
required  to  raise  unit  volume  of  a  substance  i° ;  its  dimensions 
are  L3A.     Conductivity  is  now  called  thermometric  conduc- 

k 
tivity,  or  diffusivity,  and  is  represented  by  -;  for  dimensions 

we  have — 


Remembering  that  c  =  specific  heat  x  density,  its  dimensions 
are  therefore  unity  x      .    We  obtain  at  once— 


nh  -  —  •  —  -  — 

U  J    =  LT  "*"  L3  ~  T 


ML2 

(c)  Heat  measured  as  Energy.  —  Dimensions  ~mT 

,      .  .         ML2  ML 

.*.  dimensions  of  conductivity  =  —-  4-  LTA  =        3 


The  results  of  research  on  conductivity  have  been  given 
in  varying  units,  and  therefore  it  is  necessary  to  translate  them 
into  one  set  of  units  for  purposes  of  comparison. 

Let  us  take  Forbes's  results  (p.  217).  His  units  of  length 
and  time  are  i  foot  and  i  minute. 

The  dimensions  of  [_-J  =  -~  •        i   foot  =  30-48  cm.  ;  i 
minute  =  60  seconds. 
.*.  the   multiplier,  to   change  these  numbers  into   dimisivity 

measured  in  the  C.G.S.  system,  is  ^3<      '  =  15-48 


22O  Heat 

Angstrom  gives  72*96(1  —  0*002  14^)  for  the  diffusivity  of 
copper.  Units  :  i  cm.,  i  gram,  i  minute. 

i2        i 

(1)  To  change  into  C.G.S.  system,  multiply  by  j-  =  ~r 

:.  diffusivity  =  1*216(1  —  0*002  14^) 

(2)  To  change  into  units  used  by  Forbes  (i  foot,  i  minute) 
.(i  foot  =  0*03281  cm.),  multiply  by  ---  -  -  0*00108. 

/.  diffusivity  -  72-96(1  —  0*002  14^)  x  0*00108 
=  0-078(1  —  0*002  14/) 

To  express  this  as  thermal  conductivity,  we  must  multiply 
by  c.  The  thermal  capacity  of  i  cubic  foot  of  copper  =  mass 
of  i  cubic  foot  of  copper  X  specific  heat. 

Tait  gives  the  thermal  conductivity  (/£)  of  a  specimen  of 
copper  as  4*03(1  +  0*00  13^).  Units  :  i  foot,  i  minute,  i°  C. 

The  thermal  capacity  of  i  cubic  foot  of  copper  —  c  =  mass 
of  i  cubic  foot  of  water  x  specific  gravity  of  copper  x  specific 
heat  of  copper.  Both  the  latter  quantities  will  vary  with  the 
temperature,  and  should  be  determined  for  the  bar  used  ; 
taking  ordinary  values  — 

c  -  62*51  X  8*9  X  6-0953  =  52*8 


/.  diffusivity  =  ~  g(i  +  0* 

=  0*076(1  -f-  o'ooi3/) 
To  change  this  into  C.G.S.  units,  we  have  — 

Diffusivity  =  0*076(1  +  0*001  3^)  x  15*48 
=  1*176(1  +  0-0013^) 

[Compare  Angstrom,  1*216(1  —  0-002  14/).] 
147.  Conductivity  in  Crystals.  —  Senarmont  made  a 
series  of  experiments  to  determine  whether  the  conductivity 
of  a  solid  was  the  same  in  all  directions;  this  has  been 
assumed  to  be  true  in  the  previous  experiments.  Thin 
sections  were  cut  from  crystals  in  various  directions,  a  hole 
was  drilled  in  the  centre,  and  the  sections  were  coated  with 
wax.  A  stout  copper  wire  was  passed  through  the  hole  per- 


Transmission  of  Heat 


221 


pendicularly  to  the  plane,  and  the  wire  was  then  heated  by 
a  lamp,  screens  being  arranged  so  that  the  heat  radiated  from 
the  lamp  could  not  reach  the  section  experimented  upon. 
From  the  form  of  the  area  of  wax  melted  deductions  were 
made  as  to  the  conductivity  in  various  directions.  In  the 
regular  system  of  crystals  the  part  melted  was  always  circular, 
and  it  was  inferred  that  the  conductivity  was  the  same  in 
all  directions.  In  the  other  systems  the  shape  depended  upon 
the  system  and  upon  the  direction  in  which  the  section  was 
taken  compared  with  the  principal  axis.  The  shape  was  in 


FIG.  91. 


FIG.  92. 


general  an  ellipse  (Fig.  91),  the  major  axis  being  in  the  direc- 
tion of  the  principal  axis. 

The  experiment  can  be  conducted  with  greater  care  by 
heating  the  wire  by  a  voltaic  current. 

148.  Conductivity  of  Liquids. — The  difficulty  of  pre- 
venting convection  currents  makes  the  experiments  upon  con- 
ductivity difficult  and  uncertain.  Generally,  the  conductivity 
of  liquids  is  low. 

If  the  tube  of  a  simple  air-thermometer  be  passed  through 
a  bell-jar,  as  in  Fig.  92,  and  the  bell-jar  be  nearly  filled  with 
water,  then,  by  floating  a  small  dish  containing  alcohol  on  the 
top,  arranging  that  the  dish  does  not  touch  the  sides,  we  can 
show  that  the  conductivity  of  water  is  very  small.  On  igniting 


222 


Heat 


the  alcohol,  the  steadiness  of  the  index  shows  that  no  heat 
practically  is  being  conducted  through  the  water.     If  the  bell- 
jar  be  filled  with  mercury,  and  the  experiment  be  otherwise 
repeated,  the  index  moves  at  once, 
showing  that  liquid  mercury  is,  com- 
pared with  water,  a  good  conductor. 
On  account  of  the  low  conduc- 
tivity of  water,  it  is  possible  to  heat 
to  boiling  the  water  at  the  top  of  a 
test-tube  while  ice  (sunk  by  weight- 
ing with  lead)  at  the  bottom  remains 
FIG.  93-  unmelted  (Fig.  93). 

Guthrie  used  two  cones  of  platinum  placed  base  to  base ; 
the  lower  one  (Fig.  94)  practically  formed  the  bulb  of  an  air- 


FIG.  94. 

thermometer;  water  at  a  constant  known  temperature  cir- 
culated through  the  upper  cone.  The  liquid  to  be  experi- 
mented upon  was  inserted  in  layers  of  known  thickness  between 
the  cones,  and  the  depression  of  the  liquid  in  the  graduated 
tube  of  the  air-thermometer  read. 


Transmission  of  Heat  223 

The  results  show  that,  omitting  mercury,  water  has  the 
greatest  conductivity  among  liquids,  but  that  its  conductivity 

is  only  —  that  of  copper. 
95 

CONDUCTIVITIES  OF  LIQUIDS  (k). 

Units  :  centimetre,  gram,  and  second. 

Water        ....   0-00136  Ether         ...     0*000303 

Glycerine  ...     0-000670         Alcohol     ...     0*000423 

149.  Conductivity  of  Gases.  —  If  a  piece  of  lime  be 
held  in  the  hand,  and  one  end  be  placed  in  the  flame,  the  heat 
is  conducted  to  the  hand,  and  it  is  soon  impossible  to  retain 
the  lime.  Let  the  lime,  when  cold,  be  powdered  and  placed 
on  the  palm  of  the  hand  ;  a  red-hot  poker  placed  gently  on 
the  powder,  although  a  very  short  distance  from  the  palm,  does 
not  produce  any  uncomfortable  sensation.  The  air  between 
the  particles  of  lime  form  a  good  non-conducting  surface. 

The  conductivity  of  gases  is  much  lower  than  that  of 
liquids,  and  the  difficulty  of  measuring  is  greatly  increased  by 
the  convection  currents.  Certain  results  are  obtained  from 
experiments  on  cooling,  and  also  from  the  kinetic  theory 
of  gases.  The  thermal  conductivity  of  air  is  probably  about 

that  of  copper. 


CONDUCTIVITIES  OF  GASES  (k). 

Air  ...         ...     0*000048        Carbon  dioxide    0^000040 

Oxygen        ...     0*000049        Hydrogen     ...    0*000336 
Nitrogen      ...     0*000047 

The  differences  in  the  conductivities  of  metals,  liquids,  and 
gases  make  determinations  on  conductivity  by  the  method  of 
§  143  uncertain.  Thus  if  we  take  the  conductivity  of  air  =  i, 
water  =  30  (about),  iron  =  3500,  and  copper  =20,000.  In 
measuring  the  temperature  of  the  liquid,  unless  we  get  the 
exact  temperature  at  the  interface  (this  is  generally  almost 
impossible),  we  are  measuring,  in  addition  to  the  conductivity 
of  the  metal,  that  of  a  thin  layer  of  water.  This  latter  may 


224  Heat 

disguise  altogether  the  result  due  to  the  metal  alone.  A 
similar  remark  applies  to  the  passage  of  heat  through  an  iron 
plate ;  a  thin  layer  of  air  may  offer  a  resistance  to  the  passage 
of  heat  vastly  greater  than  the  resistance  of  the  metal.  For 
this  reason  the  practical  applications  to  boilers,  etc.,  by  utilizing 
numbers  obtained  from  scientific  research,  must  be  made  with 
caution,  and  should  be  checked  with  results  obtained  from 
actual  observations.  For  practical  purposes,  the  conductivity 
of  thin  plates  of  iron  and  copper  is  regarded  as  being  perfect. 

150.  Applications. —Difference  in  conductivity  is  the 
cause  of  the  different  opinion  we  form  as  to  the  temperature 
of  bodies  (§  i)  when  we  test  them  with  the  hand. 

It  will  be  sufficient  to  give  shortly  a  few  further  illustrations. 

Water-pipes  exposed  to  the  weather  are  wrapped  in  straw, 
felt,  etc.,  to  prevent  the  loss  of  heat,  and  thus  to  avoid  the 
consequences  of  the  water  in  them  freezing.  Not  only  are  the 
straw  and  felt  bad  conductors,  but,  being  loosely  packed,  the 
air  fills  in  the  spaces,  and  it  is,  as  we  have  seen,  a  very  bad 
conductor. 

If  a  piece  of  wire  gauze  (about  9  inches  square)  be  held 
about  i  inch  above  the  gas-tap,  and  the  gas  be  then  turned 
on,  the  gas  can  be  ignited  above  the  gauze;  the  gas  below, 
however,  does  not  inflame.  The  heat  is  conducted  rapidly 
away  by  the  iron  wires  forming  the  gauze,  and  is  radiated  so 
rapidly  that  its  temperature  never  rises  sufficiently  high  to 
ignite  the  gas  that  is  in  contact  with  it  below. 

The  Davy  and  Stephenson's  "  Geordy  "  lamps  are  applica- 
tions of  this  property.  The  burning  wick  is  separated  from 
the  outer  air  by  a  closed  cylinder  of  fine  wire  gauze.  Air 
freely  enters  for  the  purposes  of  combustion,  but  the  com- 
paratively large  area  of  gauze  rapidly  conducts  the  heat  from 
any  one  part.  The  heat  is  radiated,  and  thus  the  temperature 
never  rises  high  enough  to  inflame  the  combustible  and  explo- 
sive gases  that  may  surround  it  in  a  mine.  If  the  surrounding 
gas  contains  insufficient  oxygen,  the  lamp  goes  out. 

This  applies  to  steady  burning ;  the  lamp  is  not  perfect. 
Sound-waves,  during  blasting  operations,  may  drive  the  flame 
through  the  meshes,  or  a  draught  may  direct  the  flame  on  to 


Transmission  of  Heat  225 

the  gauze,  and  raise  the  temperature  of  a  part  sufficiently 
high  to  ignite  the  dangerous  fire-damp. 

Other  examples  will  be  given  in  the  last  chapter. 

EXAMPLES.    X. 

1.  Describe  a  few  experiments  which  shall  clearly  illustrate  and  explain 
the  principle  and  application  of  the  safety-lamp. 

2.  What  is  meant  by  the  conductivity  of  a  body  for  heat  ?     Show  how 
the  conductivity  of  a  bar  of  iron  may  be  accurately  determined. 

3.  An  iron  boiler  containing  water  at  100°  C.  is  3  cm.  thick,  and  keeps  a 
room  in  which  it  is  placed  at  a  temperature  of  30°  C.  :  if  the  conductivity  of 
iron  be  1*29  unit  per  hour,  find  how  much  heat  is  given  off  per  hour  from 
a  square  centimetre  of  surface. 

4.  How  many  gram-degrees  of  heat  will  be  conducted  in  an  hour 
through  an  iron  bar  2  square  cm.  in  section  and  4  cm.  long,  its  two  extremi- 
ties being  kept  at  the  respective  temperatures  of  100°  C.  and  178°  C,  the 
mean  conductivity  of  iron  being  O'I2?    (The  units  are  a  gram,  a  centi- 
metre, a  second,  and  a  degree  Centigrade. ) 

5.  How  would  you  compare  the  thermal  conductivities  of  brass  and 
copper  ?    Two  equal  cylinders,  one  of  iron  and  the  other  of  bismuth,  are 
covered  with  wax  and  simultaneously  placed  on  end  on  a  hot  plate.     At 
first  the  melting  of  the  wax  advances  more  rapidly  on  the  bismuth  bar  ;  but 
after  it  has  melted  about  an  inch  up  both  cylinders,  the  melting  advances 
the    more    rapidly    on  the  iron  bar.     Account    for    these   phenomena. 
(London  Matric.) 

6.  Describe  Forbes's  experiments  for  determining  the  conductivity  of 
iron.     He  finds  the  conductivity  of  copper  at  200°  C.  is  0*00987.     His 
units  are  the  foot,  pound,  minute,  the  thermal  unit,  and  degree  Centigrade. 
Express  the  result  in  the  C.G.S.  units. 

7.  Explain    the    difference    between    "thermal    conductivity"    and 
"  thermal  diffusivity."     What  are  the  dimensions  of  each  ? 

8.  What  objections  have  you  to  Example  3,  based  upon  the  conductivity 
of  iron  ?     What  is  the  substance  that  is  opposing  the  passage  of  heat  ? 

9.  Show  the  difference  between  thermal  conductivity  and  diffusivity  by 
means  of  the  dimensions  of  these  two  quantities. 


226  Heat 


CHAPTER   XI. 

RADIATION. 

151.  Leslie's  Differential  Thermometer. — In  experi- 
ments  relating   to    radiant   heat   it    is    necessary   to   have   a 
thermometer  of  great  delicacy ;  the  ordinary  thermometers  are 
unsuitable.     The  investigations  of  Leslie  were  conducted  by 
the  aid  of  the  differential  thermometer  (§  13),  first  constructed 
by  him ;  one  of  the  bulbs  was  blackened  with  lampblack,  the 
better  to  absorb  the  heat.      Many  lecture  experiments  can 
be  performed  by  its  aid  j  particulars  of  some  of  these  will  be 
given.     It  is  necessary  to  see  that  the  thermometer  is  placed 
close  to  the  radiating  body. 

152.  Thermo-pile. — The    differential    thermometer  has 
been  displaced   by   the    thermo-multiplier    of   Nobili.      The 
apparatus  as  used  consists  of  two   parts — the  pile  and  the 
galvanometer. 

The  galvanometer  used  is  the  ordinary  astatic  galvanometer 
(Fig.  100)  or  the  Thomson  mirror  galvanometer. 

The  construction  of  the  thermo-pile  is  based  on  the  follow- 
ing experimental  facts  : — 

Seebeck  found  that,  if  two  bars  of  different  metals  be 
soldered  together,  and  one  of  the  junctions  be  heated,  an 
electric  current  was  produced. 

In  Fig.  95  op  is  a  bar  of  bismuth,  and  m  n  a  bent  bar 
of  copper.  The  apparatus  is  placed  in  the  magnetic  meridian. 
When  one  junction  is  heated,  the  needle  a  is  deflected  in 
such  a  manner  that  it  indicates  a  current  from  bismuth  to 
copper  at  the  hot  junction.  If  this  junction  be  cooled  by 


Radiation 


227 


melting  ice,  then  a  current  is  induced  at  the  cold  junction 
from  copper  to  bismuth.  The  current  increases  as  the  differ- 
ence between  the  temperatures  of  the  junctions  is  increased. 


FIG.  95. 

A  similar  effect  is  noticed  if  one  junction  only  be  soldered, 
while  the  other  ends  are  connected  by  a  copper  wire  (Fig.  96, 
where  B  is  a  bar  of  bismuth,  A  a  bar  of 
antimony,  and   C    the  soldered  junc- 
tion that  is  heated).     The  existence  of 
the  current  is  readily  shown  by  connect- 
ing the  unsoldered  ends  with  the  ter- 
minals of  a  galvanometer. 

The  best  effects  are  produced  by  using  antimony  and 
bismuth  as  the  two  metals.  The  current  is  increased  by  taking 
a  number  of  bars,  and  soldering  them,  as  in  Fig.  97 }  and  heating 


FIG.  97.  FIG.  98. 

the  alternate  junctions.     The  letters  have  the  same  meaning 
as  in  Fig.  96. 

In  the  thermo-pile  small  bars  of  bismuth  and  antimony  are 
soldered  together,  as  io   Fig.  98,  pieces  of  varnished  paper 


228 


Heat 


being  placed  between  to  prevent  electrical  contact.  Several 
such  sets  are  made  (four  in  Fig.  99),  and  placed  one  above  the 
other,  each  set  being  insulated  from  the  other  by  the  use  of 
varnished  paper  or  other  insulator.  In  arranging  the  sets  one 
above  another,  a  bar  of  antimony  comes  above  a  bar  of  bismuth. 
The  bismuth  b  is  then  soldered  to  the  anti- 
mony of  the  series  above  it;  the  bar  of 
this  second  series  with  the  free  end  will 
thus  be  bismuth.  It  is  soldered  to  a  bar 
of  antimony  of  the  third  series,  and  so  on. 
Thus,  when  complete,  the  only  bars  with 
a  free  end  will  be  the  "antimony  bar  a 
and  a  bismuth  bar  of  the  top  series.  The 
pile,  consisting  of  twenty  pairs  of  elements, 
is  enclosed  in  a  copper  frame,  P  (Figs.  99 
and  100).  Copper  wires  are  fastened  to  the  free  antimony 
and  bismuth  bars,  and,  suitably  insulated,  are  carried  to  the 
binding-screws  m,  n. 

The  binding-screws  are  connected  with  an  astatic  galvano- 
meter, G,  by  wires,  «,  b.    The  junctions  that  are  to  be  subjected 

n   P 


FIG.  loo. 


to  any  source  of  heat  are  usually  surrounded  by  a  cone,  C 
(Fig.  100),  in  order  to  concentrate  the  heat-rays  upon  the  face. 
In  any  experiment  we  shall  have  at  the  beginning  both  faces 
(i.e.  both  sets  of  junctions)  at  the  temperature  of  the  air,  and 


Radiation  229 

no  .movement  will  be  observed  in  the  galvanometer  needle. 
If  now  the  face  enclosed  by  C  be  directed  to  any  source  of 
heat,  the  temperature  of  that  face  will  be  above  or  below  the 
temperature  of  the  air,  according  to  the  temperature  of  the 
source  of  heat;  whether  the  temperature  is  above  or  below 
will  be  indicated  by  the  direction  in  which  the  galvanometer 
needle  moves. 

For  any  temperature  to  which  the  pile  will  be  subjected,  the 
angle  of  deflection  is  proportionate  to  the  heat  that  falls  upon 
the  pile. 

It  may  simply  be  stated  (for  fuller  details,  the  student  should 
consult  some  work  on  electricity)  that  as  heating  a  junction 
produces  a  current,  heat-energy  being  given  to  the  junction,  if 
a  current  from  a  battery  be  sent  across  the  junction  in  the 
direction  of  this  original  current  (from  bismuth  to  antimony), 
heat- energy  is  still  absorbed  and  the  junction  is  cooled.  If  by 
cooling  a  junction  a  current  is  produced,  then,  if  a  current  from 
a  battery  be  sent  in  this  direction,  the  junction  will  be  heated. 
This  is  called  the  Peltier  effect. 

153.  Radiant  Heat. — The  transference  of  heat  by  convec- 
tion and  conduction  leaves  unexplained  many  phenomena.  The 
heating  of  a  room  by  a  fire  is  not  accompanied  by  currents 
from  the  fire  to  the  sides  of  the  room  ;  and  the  interposition  of 
a  screen  between  ourselves  and  the  fire,  and  the  consequent 
shutting  off  of  the  heat,  shows  that  tfie  heat  is  not  transmitted 
by  conduction — the  particles  of  air  are  not  themselves  per- 
ceptibly heated ;  there  is  no  medium  between  the  earth  and 
the  sun  that  can  be  utilized  in  transmitting  the  sun's  heat  by 
conduction  or  by  convection ;  and  furthermore,  the  speed  with 
which  the  sun's  heat  reaches  the  earth — it  travels  with  the 
velocity  of  light — shows  that  there  is  a  different  form  of  trans- 
ference of  heat  from  the  comparatively  slow  method  of  con- 
duction or  convection.  It  has  also  been  shown  that  heat  is 
transmitted  in  vacua;  for  conduction  and  convection  some 
medium  is  necessary. 

The  identity  of  the  two  phenomena — light  and  heat — is 
further  suggested  by  the  fact  that,  after  an  eclipse  of  the  sun, 
heat  reaches  the  earth  in  the  same  time  that  light  does. 


230 


Heat 


It  is  believed  that  the  heat  of  the  sun,  the  heat  from  the 
fire  or  other  hot  body  (save  the  small  heat  that  may  be  trans- 
mitted by  convection  currents)  is  propagated  as  wave-motion. 
It  is  frequently  called  "  radiant  heat,"  but  for  the  time  being 
it  is  not  heat  at  all,  but  simply  wave-motion.  The  energy 
of  the  waves  is  converted  into  heat  when  they  impinge  upon 
any  object.  "  Radiant  energy,"  therefore,  is  a  more  accurate 
term  than  "  radiant  heat." 

154.  Wave-Motion.— Wave-motion  is  illustrated  in  the 
waves  produced  when  a  stone  is  thrown  into  a  pond ;  alternately 
we  have  crests  and  hollows  ;  the  crests  move  forward,  although 
the  particles  of  water  do  not  depart  from  their  average  position ; 
particles  of  wood,  etc.,  on  the  surface  do  not  move  forward,  but 
have  a  movement  that,  on  the  whole,  is  up  and  down.  The 
direction  of  the  wave  is  at  right  angles  to  the  motion  of  the 
water-particles.  The  waves  that  cross  a  field  when  the  wind 
blows  show  strikingly  how  a  wave  can  be  transmitted,  although 
the  particles  that  compose  the  wave  simply  move  around  a 
certain  fixed  position. 

By  giving  the  necessary  movement  to  the  end  of  a  rope,  we 
y^7^^ — w  can  send  waves  of  varying  lengths  along 

4L/\         /V        the  rope.  . 

The  idea  of  wave-motion  will  be 
aided  by  considering  the  motion  of  par- 
ticles in  a  circle.  Fasten  a  heavy 
bullet  to  a  long  fine  string,  and  set 
the  bullet  swinging  uniformly  in  a  hori- 
zontal circle  (Fig.  101,  the  plan  of  the 
circle  i,  2,  3,  ...  n,  12,  is  shown). 
Place  the  eye  in  the  same  plane  as  the 
circle,  so  that  the  motion  appears  a  straight  line  (A  B).  The 
motion  from  side  to  side  is  called  an  oscillation;  from  side 
to  side  and  back  again,  a  vibration,  that  is,  the  particle  com- 
pletes a  revolution  of  the  circle.  By  raising  or  lowering  the  eye, 
the  path  appears  an  ellipse,  and  the  same  terms  can  be  applied. 
The  diameter  (A  L  .  .  .  H  G)  of  the  circle  is  called  the 
amplitude  of  the  vibration  ;  and  the  time  of  a  complete  vibra- 
tion, the  period  of  the  vibration. 


Radiation 


231 


The  motion  in  the  line,  or  in  the  ellipse,  or  in  the  circle, 
is  called  harmonic  motion. 

Let  there  be  a  number  of  particles  moving  in  parallel 
straight  lines,  A,  B,  .  .  .  R,  S  (Fig.  102),  moving  with  harmonic 
motion,  so  that  each  succeeding  particle  begins  to  move  a 
definite  time  behind  the  preceding  one.  Suppose,  for  example, 
that  the  period  is  one  second,  and  that  each  particle  begins 

A   B    CDEFGHILKLMMOPQRS 


V 


i 


FIG.  102. 


to  move  TV  second  behind  the  preceding  particle.  Draw  the 
circle  of  given  amplitude,  and  divide  it  into  twelve  parts.  Let 
the  particles  in  the  line  A  be  at  a,  i.e.  at  4  in  the  circle,  then 
the  particle  in  line  B  will  be  -^  behind,  that  is  at  b\  the 
particle  in  C  will  be  at  c,  etc.  Join  the  points  a,  b,  c,  d,  e, 
.  .  .  and  a  sinuous  curve  is  formed  (the  thick  line).  ^ 
second,  that  is,  £  period  later,  the  particle  in  A  will  have 
moved  from  a  to  a'  (i.e.  in  the  circle-  it  will  have  moved  from  4 
to  7) ;  b  will  have  moved  to  b\  etc.  Join  these  points,  and  the 
dotted  curve  a',  V,  c\  d', .  .  .  will  be  formed ;  the  crest  has  moved 
from  E  to  H ;  the  form  of  the  wave  is  moving  from  A  to  S.  The 
position  Te^  second,  or  J  period  from  the  beginning,  is  indicated 

by  the  curve In  |f  second,  or  i  period,  the 

particles  will  be  in  their  original  positions,  but  the  crest  of  the 
wave  will  have  moved  from  E  to  Q. 

The  distance  from  crest  to  crest,  or  from  hollow  to  hollow, 
is  called  a  wave-length  ;  or,  more  accurately,  wave-length  is  the 
distance  from  any  particle  to  the  next  particle  that  is  in  a 
similar  position  in  its  path  and  is  moving  in  the  same  direction  ; 
thus  from  b  to  n  is  a  wave-length.  The  wave-length  can  also 
be  defined  as  the  distance  the  wave  travels  in  one  period. 


232  Heat 

The  student  will  see  that  similar  results  are  arrived  at  if 
the  particles,  instead  of  moving  in  lines,  moved  in  circles  or 
ellipses. 

Since  the  wave  travels  one  wave-length  in  a  period,  to 
determine  the  velocity  with  which  the  wave  travels,  it  is  only 
necessary  to  determine  the  number  of  wave-lengths  the  wave 
has  passed  over  in  a  unit  of  time. 

Wave-length  x  number  =  velocity  of  propagation 
(A)  X       (»)       -  (v)     . 

v  =  n\ 

Increasing  the  amplitude  of  the  wave  (making,  in  the  illus- 
tration, the  lines  A,  B,  C,  .  .  .  longer)  will  not  affect  the  speed 
of  the  wave  as  long  as  the  period  remains  constant.  The 
speed  depends  upon  the  period,  not  upon  the  amplitude. 

The  number  of  wave-lengths  is  evidently  the  number  that 
will  pass  any  point  in  one  second  (if  i  second  be  the  unit  of 
time) ;  n,  therefore,  which  measures  the  number  of  vibrations 
per  second,  also  measures  the  frequency,  or  the  number  of 
times  a  wave  will  affect  a  stationary  object  in  its  path  in  a 
unit  of  time. 

The  speed  with  which  radiant  .energy  is  transmitted  varies 
with  the  density  of  the  medium ;  it  is  less,  for  example,  in  air 
than  in  vacno^  less  in  glass  than  in  air;  but  the  number  of 
vibrations  keeps  constant,  therefore  the  wave-length  varies  with 
the  speed.  Wave-lengths,  then,  must  always  be  referred  to 
some  standard  subject  (air)  under  given  conditions  or  to  vacua. 
The  quantity  that  does  not  change  is  the  frequency  (#). 

155.  Light  and  Heat. — It  is  believed  that  radiant  energy 
is  propagated  by  waves  in  which  the  particles  of  ether  that  fill 
all  space  move  at  right  angles  to  the  direction  in  which  the 
waves  move;  this  allows  of  great  variety  in  movement — the 
particles  may  move  in  straight  lines  vertically  or  horizontally, 
or  partake  of  both  motions,  or  may  move  in  circles  or  ellipses. 

This  is  a  brief  but  by  no  means  complete  statement  of  the 
method  by  which  radiant  energy  is  propagated;  the  whole 
process  is  probably  much  more  complex  than  the  illustration 
deduced  from  the  oscillations  of  a  single  row  of  particles,  and 


Radiation  233 

the  complete  phenomenon  is  further  complicated  by  the  fact 
that  waves  of  varying  lengths  are  being  propagated  at  the  same 
time.  For  a  fuller  statement,  as  well  as  for  the  methods  by 
which  the  wave-lengths  may  be  estimated,  some  book  on 
optics  should  be  consulted. 

The  heat  of  a  body  is  due  to  the  kinetic  energy  of  its 
molecules.  Part  of  this  energy  is  used  in  setting  in  motion 
the  ether-particles  that  by  wave-motion  transmit  the  energy  to 
other  bodies ;  in  the  process  of  transmission  it  is  not  heat,  but 
simply  a  form  of  energy.  The  waves,  on  meeting  a  substance, 
may  be  retarded  or  destroyed.  The  energy  of  the  wave  may 
then  produce  certain  effects  :  (i)  it  may  increase  the  vibrations 
of  the  molecules  of  the  new  substance,  that  is,  increase  the 
kinetic  energy,  and  therefore  raise  its  temperature,  or,  if  the 
substance  be  on  the  point  of  changing  into  the  liquid  or  vapour 
state,  the  radiant  energy  may  assist  in  the  process,  and  thus 
increase  the  potential  energy— in  both  cases  acting  as  if  heat 
were  given  to  the  new  substance ;  (2)  falling  upon  a  suitable 
substance,  it  may  produce  the  effect  of  light,  or,  falling  upon 
our  eyes,  it  may  produce  the  sensation  of  light ;  or  (3)  the 
radiant  energy  may  be  able  to  decompose  certain  chemical  com- 
pounds such  as  silver  chloride.  These  effects  are  not  sharply 
separated,  and  the  total  result  of  the  radiant  energy  may  be 
compounded  of  all  three,  that  is,  of  heat,  light,  and  chemical 
effects. 

The  wave-lengths  are  very  small,  and  are  usually  expressed 
in  millionths  of  a  millimetre  (o'oooooi  =  i  mm.  -f-  io6  = 
i  mm.  x  io~6,  written  i/x/x).  For  the  methods  by  which 
these  measurements  are  made,  some  work  on  optics  must  be 
consulted.  On  account  of  its  importance  we  may  repeat 
that  the  velocity  of  radiant  energy  is  lessened  in  passing 
through  denser  media,  the  wave-length  becoming  proportion- 
ately less.  The  number  of  vibrations  or  the  frequency  (n) 
remains  constant.  In  defining  a  particular  wave,  either  the 
frequency  or  the  wave-length  in  a  stated  medium  (air,  for 
example)  under  given  conditions  as  to  temperature  and  pressure 
must  be  given. 

The  velocity  in  vacuo  is,  according  to  the  best  determina- 


234  Heat 

tion,  186,300  miles  per  second;  this  is  nearly  300,000  kilo- 
metres per  second  =  3  X  io10  centimetres  per  second. 

The  wave-lengths  that  have  been  measured  lie  between 
i5,ooo//,/x,  and  290^;^. 

All  bodies  that  we  are  acquainted  with  are  at  some  tempe- 
rature ;  their  molecules  are  therefore  in  a  state  of  motion,  and 
are  able  to  start  waves  that  transmit  the  radiant  energy.  The 
coldest  bodies  will  therefore  originate  ether-waves  that  are 
comparatively  long  waves.  The  present  methods  of  research 
can  only  detect  waves,  however,  when  their  lengths  are  less 
than  i5,ooo/x,/x.  Until  the  last  few  years  all  waves  shorter 
than  39o/x,/x  and  longer  than  770^  were  unknown.  As  the 
temperature  of  a  body  rises,  shorter  waves  are  added  to  the 
long  waves  already  existing,  and  the  combined  effect,  when 
these  waves  fall  on  a  suitable  object*,  is  heat.  When  waves 
27OO/Z/X  in  length  are  added,  we  are  dealing  with  waves  that 
we  can  detect  in  the  radiant  energy  we  receive  from  the  sun. 
If  we  imagine  that  the  temperature  of  a  hot  body  continues 
to  rise,  waves  whose  lengths  are  774^  begin  (there  is  no 
sudden  change  in  the  lengths,  the  shortening  is  gradual),  the 
total  effect  being,  up  to  the  present,  that  the  radiant  energy, 
when  destroyed,  is  changed  into  -heat.  Waves  shorter  than 
774/>t/x  produce  a  further  phenomenon.  Not  only  do  they 
produce  heat,  but  they  affect  the  eye  and  produce  the  sensation 
of  light,  and  the  body  which,  up  to  the  present,  would  in  a 
dark  room  have  been  invisible,  now  glows  with  a  dark  red 
colour.  As  the  temperature  further  rises,  more  heat  is  received, 
and  it  appears  to  the  eye  bright  red,  and  ultimately  it  reaches 
a  "  white  heat,"  or  is  incandescent :  the  last  waves  started 
are  about  440^  long  ;  they  have  not  greatly  increased  the  heat 
transmitted.  A  further  rise  in  temperature  starts  yet  shorter 
waves,  but  they  add  nothing  to  the  heat  transmitted  as  far  as 
our  instruments  are  able  to  measure,  and  they  add  nothing 
further  to  the  light.  Before  incandescence  was  reached,  the 
shorter  waves  were  able  to  decompose  silver  chloride;  the 
waves  added  after  incandescence  are  peculiarly  effective  in 
this  respect,  and  their  effects  can  therefore  be  photographed 
by  using  silver  chloride  paper.  We  can  by  such  means 


Radiation  235 

determine  the  existence  of  waves  whose  lengths  are  295/1^. 
Doubtless  there  are  others  whose  lengths  are  yet  less,  but  at 
present  we  have  no  means  of  detecting  them. 

Up  to  the  present  we  have  taken  the  combined  effect  of 
the  waves.  In  §  167  it  will  be  shown  how  these  waves  can  be 
separated. 

The  important  matter  for  the  student  to  remember  is  that 
light,  heat,  and  chemical  action  are  effects  due  to  one  pheno- 
menon— the  transformation  of  the  radiant  energy  of  ether- 
waves  ;  the  effects  vary  with  the  lengths  of  the  waves.  It 
should  be  added  that,  in  the  above  explanation,  only  the 
earlier  known  chemical  effects  are  taken;  with  other  sub- 
stances than  silver  chloride,  some  of  the  shorter  waves  pro- 
duce chemical  effects.  In  heat  we  are  concerned  with  the 
waves  that  produce  the  effect  of  heat;  these,  as  is  known 
at  present,  lie  between  i5,ooo/x/x  and  290^.  The  greatest 
heating  effects  are  produced  by  waves  whose  lengths  are 
about  noo^t/x — waves  unable  to  produce  the  sensation  of 
light. 

In  the  next  few  paragraphs  we  shall  examine  the  total 
heat  effects  of  radiant  energy  from  any  source ;  but  most  of  the 
results  will  equally  well  apply  to  the 
heating  effects  of  any  particular 
waves. 

156.  Law  of  Inverse  Squares] 
— Radiant  energy,  in  a  homogeneous 
medium,  travels  in  a  straight  line ; 
and  as  a  result  of  rectilinear  motion 
we  have  the  law  that — 

The  heating  effect  varies  inversely 

,  °    ,       ,.  FIG.  103 

as  the  square  of  the  distance. 

If  the  source  of  heat  at  the  centre,  c  (Fig.  103),  of  a  sphere 
be  very  small,  and  supply  an  amount  of  heat  whose  measure  is  Q, 
then,  the  area  of  the  sphere  being  ^(caf^  the  quantity  received 

on  unit  area  is  — — -  If  the  heat  falls  on  a  sphere  of 
radius  ce,  the  quantity  received  on  unit  area  is  — ~— 


Heat 


i.e.  the  amount  received   per  unit  area  on  the  sphere  a  b  : 
amount  received  on  the  sphere  ef  — 

Q    .  _Q_  =  j_  .  _L 

This  is  experimentally  verified  by  Melloni's  apparatus. 

A  tin-plate  box,  M  (Fig.  104),  is  filled  with  hot  water,  and 
is  kept  at  a  constant  temperature;  one  side  is  coated  with 


FIG.  104. 

lampblack,  and  is  directed  towards  the  thermo-pile  fitted  with 
its  conical  reflector,  and  the  deflection  is  noted  at  any  given 
distance.  The  thermo-pile  is  now  moved  away,  so  that  the 
base  of  the  cone,  a  £,  is  enlarged  to  A  B  (Fig.  105).  As  long  as 
the  base  of  the  cone  falls  on  the  lampblack  surface,  there  is 
no  change  in  the  deflection,  showing  that  the  quantity  of  heat 
received  per  unit  area  in  each  case  is  the  same. 

The  cones  ca  b  and  C  A  B  are  similar.  Since  the  temperature 
of  the  box  is  kept  constant,  the  quantity  of  heat  emitted  will 
be  as  the  areas  of  the  circles  a  b  and  A  B ;  that  is,  as  (oaf  is 
to  (OA)2,  or  (because  the  cones  cab  and  CA  B  are  similar)  as 
(cof  is  to  (CO)2.  Since  the  quantities  emitted  are  as  the 
squares  of  the  distances,  and  the  heating  effect  on  the  thermo- 
pile is  the  same,  the  heat  that  falls  on  the  thermo-pile  must 
vary  inversely  as  the  square  of  the  distance. 


Radiation  237 

If  the  position  of  the  galvanometer  needle  be  noted  when 
the  box  is  in  the  position  of  Fig.  105,  so  that  C  A  B  is  a  right 


FIG.  105. 

cone,  and  if  the  box  be  then  slightly  turned  on  a  vertical  axis 
through  O,  so  that  the  base  of  the  imaginary  cone  is  still  on  the 
face  of  the  box,  then  no  change  is  made  in  the  position  of  the 
needle. 

157.  Intensity. — The   physical  meaning  of   change  of 
"  intensity  "  for  heat-rays,  as  long  as  their  period  remains  con- 
stant, is  that  the  amplitude  of  vibration  has  been  changed.    By 
doubling  the  amplitude,  the  intensity  is  quadrupled;  by  halving 
the  amplitude,  the  intensity  is  reduced  one-fourth ;  so  that  the 
intensity  of  heat  varies  as  the  square  of  the  amplitude. 

It  has  already  been  stated  that  radiant  energy  is  transmitted 
by  means  of  the  vibration  of  the  ether-particles.  If  we  could 
see  the  motion  of  these  particles,  we  should  observe  the  greatest 
amplitude  near  the  source  of  the  radiant  energy  (a  hot  ball  kept 
at  a  constant  temperature,  let  us  suppose) ;  as  we  moved  from 
the  source  of  heat,  the  amplitude  would  decrease,  the  wave- 
length remaining  constant.  At  double  a  given  distance,  the 
amplitude  would  be  halved ;  at  three  times  the  distance,  the 
amplitude  would  fall  to  one-third  of  the  original  amplitude. 

158.  Radiant  Energy  propagated  in  Vaeuo.— The 
transmission  of  radiant  energy  is  independent  of  air-particles, 


238  Heat 

and  is  due  to  the  vibrations  of  ether-particles.  That  radiant 
energy  can  be  transmitted  through  a  vacuum  is  shown  by  an 
experiment  due  to  Rumford. 

A  thermometer  tube,  /  (Fig.  106),  is  sealed  into  a  glass  flask 
so  that  the  bulb  is  at  the  centre.  After  softening  and  then 
narrowing  the  neck  of  the  flask,  a  vacuum  is 
produced  by  suitable  means,  and  the  tube  is 
sealed.  On  immersing  the  flask  in  warm' water, 
or  bringing  any  hot  body  near,  the  thermometer 
at  once  rises.  The  effect  could  not  be  produced 
by  heat  being  transmitted  from  the  air  by  means 
of  the  glass  of  the  flask  or  thermometer  tube ;  it 
is  transmitted  by  radiation  through  the  vacuum  in 
the  interior. 

The  transmission  from  the  incandescent  car- 
bons or  platinum  filaments  of  an  incandescent 
lamp,  and  the  effect  of  a  hot  body  on  the  waves 
of  Crookes's  radiometer  (§  173),  afford  other 
illustrations  of  the  fact  that  heat  is  transmitted  in  vacua ;  and 
further,  the  fact  that  radiant  energy  from  the  sun  reaches  us 
can  only  be  explained  by  assuming  that  air-particles  are  not 
necessary  for  the  transmission  of  radiant  energy. 

159.  Reflection. — If  the  waves  by  which  radiant  energy 
is  transmitted  meet  a  suitable  surface,  part  are  reflected.  The 
laws  that  govern  the  reflection  are — 

(i)  The  incident  ray,  the  reflected  ray,  and  the  normal  of 
the  surface  are  in  the  same  plane. 

(2)  The  angle  of  incidence  is 
equal  to  the  angle  of  reflection. 

Let  m  n  (Fig.  107)  be  a  plane 
reflecting  surface,  B  D  the  normal 

FiGBio7  *  at  the  point  B >  then  if  the  ray  of 

heat  (or  light)  strike  the  surface  in 

the  direction  C  B,  and  be  reflected  as  B  A,  C  B  is  called  the 
incident  ray,  B  A  the  reflected  ray ;  the  angle  C  B  D  is  the 
angle  of  incidence  (/),  and  DBA  the  angle  of  reflection  (r). 
CB,  B-D,  B  A  are  in  one  plane,  and  the  angle  C  B  D  equals 
the  angle  D  B  A. 


Radiation  239 

The  laws  as  regards  the  waves'  that  cause  the  sensation 
of  light  are  readily  illustrated  by  mirrors ;  for  heat,  simple 
modifications  of  such  experiments  are  needed.  The  dif- 
ferential thermometer  is  sufficiently  sensitive  for  rough 
experiments. 

One  bulb  of  the  differential  thermometer  T  (Fig.  108)  is 
blackened  to  render  it  more  sensitive.  A  semicircle,  about 
2\  feet  radius,  is  drawn  on 
a  table  and  graduated,  and 
at  the  centre  a  sheet  of  tin, 
M,  is  placed  vertically. 
Two  tin  tubes  are  directed 
towards  M ;  at  the  end  of 
one  an  iron  ball  at  dull  heat 
(C)  is  placed,  at  the  end 
of  the  other  the  blackened 
bulb  of  the  differential  thermometer.  The  ball  and  its  tube 
being  fixed,  the  other,  always  directed  towards  M,  is  moved 
until  the  thermometer  is  affected. 

Whatever  angle  one  tin  makes  with  the  normal  M  o,  the 
other  makes  an  equal  angle,  and  as  all  are  in  the  plane  of  the 
table,  the  first  two  laws  are  demonstrated. 

By  keeping  the  source  of  heat  constant  (keep  the  ball  in 
the  flame  of  a  large  Bunsen  burner),  and  substituting  polished 
plates  of  brass,  iron,  similar  roughened  plates,  and  plates 
covered  with  lampblack,  etc.,  at  M,  and  noting  the  position  of 
the  index  of  the  thermometer,  we  can  roughly  compare  the 
reflecting  powers  of  various  materials.  By  such  experiments 
substances  have  been  arranged  according  to  their  comparative 
powers  of  reflecting  heat. 

Remembering  that  the  finding  of  the  position  of  the  foci 
of  concave  mirrors  is  based  on  the  law?  of  reflection,  we  can 
further  verify  the  laws  as  regard  radiant  energy. 

The  foci  of  two  concave  mirrors  are  determined,  and  the 
mirrors  are  arranged  facing  each  other  (Fig.  109).  At  one  focus, 
n,  is  placed  a  hot  copper  ball ;  a  piece  of  phosphorus  placed 
at  the  other  focus,  m,  is  at  once  ignited,  or  a  small  flask  of 
water  in  the  same  position  can  be  warmed.  It  can  be  shown, 


240 


Heat 


by  protecting  m  from  the  direct  rays  of  «,  that  the  heating 
effect  is  not  produced  by  direct  radiation. 

If  one  of  these  mirrors  be  directed  towards  the  sun,  pieces 
of  wood  or  paper  are  ignited  at  the  focus. 


FIG.  109. 


These  experiments  have  been  repeated  by  Davy  in  vacuo, 
thus  illustrating  the  statement  made  in  §  159. 

If  a  flask  containing  melting  ice  or  other  cold  body  be 
placed  at  ;;/,  a  thermometer  placed  at  n  falls  in  temperature. 
This  apparent  reflection  of  cold  will  be  dealt  with  later. 


FIG.  no. 


Suggestions  have  already  been  made  for  testing  the  reflect- 
ing powers  of  bodies.  Leslie  used  a  cube  of  boiling  water,  A, 
as  a  source  of  constant  heat  (Fig.  1 10).  The  rays,  after  reflection 


Radiation  241 

at  the  concave  mirror  M,  were  received  upon  a  small  plane 
mirror  of  polished  metal  placed  nearer  to  the  mirror  than  the 
focus  F  j  the  heat-rays  were  thereby  reflected  and  brought  to 
a  focus  upon  one  bulb  of  the  differential  thermometer.  By 
using  different  plane  reflectors  all  of  equal  size,  Leslie  was  able 
to  assert  that  each  mirror  received  equal  quantities  of  heat, 
and  that  the  varying  effects  upon  the  differential  thermometer 
were  due  to  the  differences  in  the  reflecting  power  of  the 
substances  used.  By  such  means  he  obtained  the  comparative 
reflecting  powers  of  various  substances,  making  polished  brass, 
which  he  regarded  as  the  best  reflector,  equal  to  100. 

Polished  brass    ...  100  Indian  ink            ...  13 

Silver      90  Glass        10 

Steel       70  Oiled  glass           ...  5 

Lead       ...         ...  60  Lampblack          ...  o 

Desains  and  De  la  Provostaye  used  Melloni's  apparatus 
(Fig.  112).  First  the  effect  of  a  direct  ray  from  a  lamp  was 
observed  by  noting  the  deflection  produced  in  the  galvano- 
meter connected  with  the  thermo-pile ;  then  the  effect  was 
measured  when  the  ray  was  reflected  from  various  substances  ; 
in  this  way  the  ratio  of  the  intensity  of  the  direct  ray  to  the 
intensity  of  a  reflected  ray  was  determined. 

The  reflecting  power  varied  with  the  source  of  heat  and 
varied  also  with  the  angle  of  incidence.  When  the  source  of 
heat  was  a  Locatelli  lamp,  and  the  angle  of  reflection  50°,  the 
following  numbers  were  obtained  : — 

Silver  plate        ...  0-97  Steel      0-82 

Gold      0-95  Zinc       o'8i 

Brass      0-93  Iron       077 

Platinum            ...  0*83  Cast  iron           ...  074 

160.  Diffusion  of  Radiant  Energy. — If  a  ray  of  light  in 
an  othenvise  dark  room  fell  upon  a  piece  of  polished  metal  or 
a  good  mirror,  the  light  is  reflected  regularly,  and  the  surface 
can  only  be  seen  if  the  eye  be  placed  in  the  line  of  the  reflected 
ray.  If  the  surface  be  rough — if,  for  example,  the  mirror  be 
dusty,  or  a  piece  of  paper  take  its  place,  then  part  of  the  light 

R 


242  Heat 

is  reflected  irregularly,  or  is  diffused,  and  the  surface  can  be 
seen  from  any  portion  of  the  room.  An  analogous  phenomena 
takes  place  with  the  obscure  heat-rays.  The  thermo-pile  is  only 
affected  in  the  case  of  perfect  reflectors  when  it  is  in  the 
path  of  the  reflected  ray.  If  the  ray  falls  upon  a  surface  like 
white  lead  or  Indian  ink,  then  the  thermo-pile  is  affected  in 
whatever  position  it  is  placed  in  front  of  such  surfaces  ;  the 
effect  is  obtained  as  soon  as  the  ray  falls  upon  the  surface,  and 
does  not  increase  as  the  surface  is  heated,  as  it  would  do  if  the 
result  were  due  to  the  heating  of  the  surface  and  the  consequent 
radiation. 

The  diffusive  power  of  lampblack  is  practically  nil,  as  is 
its  reflecting  power ;  the  whole  of  the  radiant  energy  that  falls 
upon  it  is  absorbed  and  radiated.  The  diffusive  power  varies 
with  the  source  of  heat ;  white  lead,  for  example,  has  a  marked 
diffusive  power  if  the  source  of  heat  be  at  a  high  temperature, 
that  is,  if  exposed  to  sun-heat  or  to  the  heat  from  incandescent 
platinum;  when  the  source  of  heat  is  copper  at  100°  C.,  practi- 
cally all  is  absorbed  and  none  diffused.  When  sunlight  falls 
upon  ordinary  objects,  only  certain  light-rays  are  diffused ;  these 
produce  the  sensation  of  colour ;  the  other  rays  are  absorbed. 
Analogously,  when  sun-heat  falls  upon  objects,  certain  of  these 
rays  are  absorbed ;  the  others  are  diffused,  the  rays  diffused 
differing  in  various  substances. 

161.  Emission,  Absorption,  Transmission.— The 
general  facts  regarding  emission,  absorption,  and  transmission 
are  readily  demonstrated. 

If  the  cube  (Fig.  1 10)  be  kept  rilled  with  hot  water,  and  three 
of  the  vertical  sides  be  coated  with  (i)  lampblack,  (2)  paper, 
(3)  isinglass,  and  (4)  one  side  be  kept  polished,  then,  by  bring- 
ing the  hand  in  turn  near  each  side,  we  can  determine  that 
the  above  order  is  the  order  of  the  powers  of  the  substances 
as  radiators.  A  bulb  of  a  differential  thermometer,  blackened 
with  lampblack  (to  render  it  a  better  absorber),  shows  the 
result  more  decisively ;  while  the  delicate  thermo-pile  answers 
yet  better.  Whatever  be  the  measurer  used,  it  should  be  held 
in  turn  at  equal  distances  from  each  side. 

A  series  of  similar  tin   or   brass  vessels,  coated  on  the 


Radiation  243 

outside  with  different  materials,  may  be  filled  once  for  all  with 
equal  quantities  of  hot  water,  and  left  to  cool.  By  reading 
the  thermometers  placed  in  each  vessel  at  equal  times,  we 
can  determine  which  have  lost  the  greatest  amount  of  heat, 
and  thus  determine  which  substances  radiate  the  more  readily. 
Lampblack,  and  similar  rough  black  substances,  are  the  best 
radiators ;  to  a  less  degree  we  have  paper  and  roughened  metals 
(by  mechanical  roughing  or  by  oxidation),  while  polished 
metals  are  the  worst  radiators. 

Place  the  vessels,  with  their  bases  covered  with  the  sub- 
stances to  be  examined,  above  a  hot  plate,  kept  at  a  constant 
temperature,  or  place  them  with  their  sides  covered,  at  equal 
distances  from  the  same  source  of  heat  (for  rough  purposes, 
at  equal  distances  from  a  fire),  fill  the  vessels  with  cold 
water,  and  examine  the  thermometers  at  the  end  of  five 
minutes. 

It  will  be  found  that  the  vessel  coated  with  lampblack 
shows  the  greatest  rise  in  temperature ;  next  come  those  with 
surfaces  like  paper ;  and  last,  and  worst,  those  whose  surfaces 
are  polished  metals. 

The  substances,  arranged  according  to  their  power  of 
absorbing  radiant  energy,  agree  with  their  order  as  radiators. 

It  follows,  therefore,  that  water  boils  the  quicker  in 
blackened  kettles,  while  such  a  kettle,  if  removed  from  the 
fire,  soon  loses  its  heat.  To  keep  water  hot,  it  must  be  kept 
in  polished  vessels. 

We  have  seen  that  lampblack,  paper,  rough  metals,  polished 
metals,  are  arranged  in  the  reverse  order  of  their  reflecting 
powers,  and  we  come  to  the  generalization — 

Good  reflectors  are  bad  radiators  and  bad  absorbers. 

Bad  reflectors  are  good  radiators  and  good  absorbers. 

The  experiments  of  Leslie  can  also  be  performed  by  the 
student  (Fig.  in).  In  comparing  the  radiating  power,  he 
kept  his  cube  filled  with  hot  water,  and  presented  the  faces 
covered  with  the  substances  in  turn  to  a  spherical  mirror,  M. 
His  thermometer  was  placed  in  the  focus  F,  and  from  the 
indications  he  constructed  his  table  of  the  comparative  powers 
x>f  substances  as  radiators. 


244 


Heat 


Melloni's  apparatus  being  more  delicate,  the  mirror  was 
unnecessary.  The  rays  passed  through  a  hole  in  a  screen 
(Fig.  112),  and  fell  directly  upon  the  thermo-pile. 

In  order  to  measure  the  absorbing  power,  he  kept  a  con- 
stant source  of  heat,  and  allowed  the  rays  to  fall  upon  the 


FIG. 


face  of  a  copper  disc  covered  with  the  given  substance  :  the 
other  face  was  covered  with  lampblack,  and  beyond  this  was 
the  thermo-pile.  He  found  that  the  radiating  power  and 
absorbing  power  of  a  thick  layer  "of  lampblack  was  practically 


perfect.      He   called 
following  tables : — 


this   power   100,   and   constructed   the 


Emissive  po*ver.                                           Absorbing  power. 
Temperature  100°  C.                                    Source  of  heat  100°  C. 

Lampblack 
White  lead 

IOO 
IOO 

Lampblack 
White  lead 

IOO 
IOO 

Isinglass 
Indian  ink 
Shellac  
Polished  metal   3 

85 

72 

to  17 

Isinglass 
Indian  ink 
Shellac  
Polished  metal  .  .  . 

85 

72 

13 

162.  Effect  of  the  Source  of  Heat  on  Absorption.— 

The   experiments  were  repeated  by  varying  the   source   of 
heat.     Lampblack  is  taken  as  100  at  all  temperatures. 
The  results  are  contained  in  the  following  table  : — 


Radiation 


245 


Source  of  heat. 


Substance. 

. 

Locatelli's 
lamp. 

Incandescent 
platinum. 

Heated 
copper. 

Hot-water 
cube. 

Lampblack    .  . 

100 

100 

100 

100 

Indian  ink 

96 

95 

87 

85 

White  lead     .. 

53 

56 

89 

100 

Isinglass 

52 

54 

64 

91 

Shellac 

48 

47 

70 

72 

Metallic  surface 

14 

I3-5 

13 

13 

At  high  temperatures,  Indian  ink  absorbs  more  than  white 
lead,  while  at  the  temperature  of  boiling  water  white  lead  is 
equal  to  lampblack,  while  Indian  ink  has  fallen  from  96  to  85. 

163.  Diathermancy. — Just  as  substances  like  glass, 
crystal,  sheets  of  paper,  allow  a  certain  amount  of  light  to  pass 
through,  and  are  called  transparent  or  translucent,  while  other 
substances,  as  metals  (save  when  very  thin),  are  opaque,  so 
substances  like  glass,  crystal,  allow  heat  rays  to  be  readily  trans- 


FlG     112. 


mitted,  and  are  diathermanous ;  the  term  "  adiathermanous  " 
being  used  for  bodies  that  do  not  transmit  heat. 

164.  Transmission. — In  the  Melloni  apparatus  (Fig.  112) 
the  rays  from  a,  the  source  of  heat,  pass  through  the  hole  in  the 


246 


Heat 


screen  E,  and,  after  passing  through  a  thin  plate  of  the  material, 
C,  fall  upon  the  thermo-pile  A  B  ;;/.  The  deflection  of  the 
galvanometer  D  is  noted  first  before  C  is  placed  on  the  stand. 
This  measures  the  total  radiation  (d)  from  the  given  source 
of  heat.  When  C  is  interposed,  the  deflection  (d')  is  again 
measured.  Then  the  percentage  of  rays  transmitted  equals 


Liquids  are  enclosed  in  a  trough  whose  sides  are  made  of 
rock  salt.  The  deflection  is  first  measured  when  the  trough  is 
empty,  then  when  it  contains  the  liquid  ;  the  difference  in  the 
deflection  measures  the  amount  absorbed  by  the  liquid,  and  by 
subtracting  this  from  100,  the  percentage  of  rays  transmitted 
is  obtained. 

Gases  were  enclosed  by  Tyndall  in  a  cylinder,  A  (Fig.  113), 
about  4  feet  long,  fixed  horizontally,  whose  ends  were  closed 


FIG  113. 

by  rock-salt  plates.  By  means  of  the  tube  r,  that  was  joined 
to  a  suitable  pump,  the  cylinder  could  be  exhausted,  and  the 
gas  could  then  be  admitted  by  the  tube  /.  The  thermo-pile 
T,  and  its  galvanometer  M,  were  at  one  end,  and  the  source 
of  heat  C  at  the  other.  To  increase  the  delicacy  of  his 
apparatus,  in  order  to  measure  the  very  small  percentage  of 
absorption,  another  similar  source  of  heat,  D,  was  placed 
opposite  the  other  end  of  the  thermo-pile.  By  the  aid  of  a 
screen,  it  could  be  so  arranged  that  the  effects  of  C  and  D 
were  such  that  the  galvanometer  needle  stood  at  zero  when 
the  cylinder  was  exhausted.  The  admission  of  a  gas  or  vapour 
disturbed  this  equilibrium  and  the  deflection,  because  heat  was 
absorbed  by  the  vapour;  the  radiation  from  the  cube  D  had  now 
the  greater  effect,  and  the  deflection  measured  the  absorption. 


Radiation 


247 


165.  Melloni's  Sources  of  Heat.— Melloni  varied  his 
experiments  by  using  four  sources  of  heat.  These  were  (Fig. 
114) :  (i)  A  Locatelli  lamp  with  a  compact  wick,  no  chimney- 
glass,  and  provided  with  a  reflector,  L;  (2)  a  spiral 


of 


FIG.  114. 

platinum,  S,  kept  incandescent  in  the  flame  of  a  spirit-lamp ; 
(3)  a  blackened  copper  plate,  kept  at  a  constant  temperature 
(about  400°  C.)  by  a  spirit-lamp,  C ;  and  (4)  a  copper  tube 
blackened  with  lampblack,  E  (kept  at  100°  C). 

166.  Refraction. — When  radiant  energy  meets  a  surface, 
part  is  reflected ;  another  portion  may  enter  the  new  medium, 
set  in  motion  the  ether-particles  of  the  medium,  and  the  wave 
continues  its  course.  If  the  surface  be  perpendicular  to  the 
wave-front,  the  speed  of  the  wave  is  diminished  when  it  enters 
a  denser  medium ;  light,  for  example  (the  waves  that  affect  the 
eye),  travels  with  diminished  speed  through  glass.  If  the  wave- 
front  makes  an  angle  with  the  surface,  as  a  consequence  of 
the  lessened  speed,  it  swings  round,  so  that  the  angle  the 
direction  of  the  wave  makes  with  the  normal  is  lessened. 
This  phenomenon,  known  as  refraction,  is  readily  illustrated 
in  the  case  of  the  waves  that  produce  the  effect  of  light.  The 
beam  of  light,  after  reflection  from  the  mirror  M  (Fig.  115),  meets 
the  surface  of  the  water  at  C,  making  the  angle  B  C  N  with  the 
normal ;  part  is  reflected  (C  R),  and  part  passes  into  the  water  as 
the  beam  C  E ;  the  angle  E  C  F  is  less  than  the  angle  B  C  N. 


248 


Heat 


The   ratio 


sine  BCN 


sine  ECF 
constant  for  the  two  media. 


is   the   index   of  refraction,  n,  and  is 


FIG. 115. 


In  the  case  of  air  and  water,  n  =  -. 

o 

This  expresses  the  fact 
that  the  speed  of  light  in 
air  is  to  its  speed  in  water 
1:4:3.  This  explana- 
tion applies  strictly  to 
waves  of  a  definite 
length;  but  the  disper- 
sion is  so  slight  in  water 
that  the  ray  of  light  may 
be  considered  as  a  whole. 
An  analogous  pheno- 
menon is  observable  in 
the  case  of  heat-waves. 

The  shorter  the 
waves,  the  greater  the  refraction.  Therefore  the  waves  that 
produce  the  sensation  of  violet  will  be,  after  refraction,  further 
from  the  normal  than  those  that  produce  the  sensation  of  red, 
and  consequently  a  beam  of  white  light  is  dispersed.  (In  the 
case  of  air  and  water  the  dispersion  is  so  slight  that  it  is  diffi- 
cult to  observe.) 

As  a  result  of  refraction,  if  the  denser  medium  be  in  the 
form  of  a  prism,  after  the  beam  has  passed  through  it  is  found 
that  its  direction  has  been  turned  towards  the  base  of  the 
prism  (see  Fig.  116) — the  shorter  waves  show  the  change  of 
direction  more  than  the  longer  ones. 

167.  The  Spectrum. — This  allows  us  to  separate  the 
waves.  Light  from  a  slit,  S  (Fig.  116),  is  focussed  by  the  lens 
A  on  to  the  prism  B.  If-  the  light,  after  emerging  from  the 
prism,  be  received  upon  a  screen,  the  coloured  spectrum  is 
produced ;  the  violet  rays,  V,  are  refracted  the  most,  the  red,  R, 
the  least.  The  spectrum  from  violet  to  red  is  visible  to  the  eye ; 
these  colours  are  caused  by  the  light-waves.  The  existence 
of  rays  refracted  more  than  the  violet  (the  actinic  rays)  is  made 
evident  by  the  effect  on  photographic  paper,  and  the  existence 


Radiation 


249 


of  longer  waves  beyond  the  red,  the  so-called  dark  rays  (from 
R  to  O),  is  proved  by  the  use  of  the  thermo-pile  /. 

A  thermo-pile  whose  face  was  made  of  a  number  of  elements 
in  a  line,  was  used  in  Tyndall's  researches.      Its  face  was 


FIG.  116. 

protected  by  a  slit  that  could  be  made  as  narrow  as  was 
required.  This  face  was  moved  along  the  spectrum,  and  the 
effect  observed  on  a  delicate  galvanometer.  The  results  were 
recorded  as  in  Fig.  117.  b  (blue),  g  (green),  y  (yellow), 
o  (orange),  r  (red),  represent  the  visible  spectrum  from  E  to  D. 


FIG.  117. 

Vertical  lines  are  raised  proportional  to  the  deflection  indicated 
by  the  galvanometer,  and  a  curve  drawn  joining  the  tops  of 
these  vertical  lines. 

The  source  of  radiant  energy  was  that  of  the  electric  arc, 


250 


Heat 


the  lens  and  prism"  were  of  rock  salt,  a  substance  that  does  not 
absorb  the  heat-rays.  Beyond  the  blue  (slightly)  there  is  no 
heating  effect ;  there  is  a  gradual  increase  towards  the  red ;  but 
beyond  the  red,  in  a  part  that  produces  no  effect  of  colour, 
we  have  a  marked  increase.  A  maximum  is  reached  (B)  at  a 
distance  beyond  the  red  equal  to  one-half  of  the  visible  spec- 
trum. There  is  then  a  decrease,  the  instrument  ceasing  to  be 
affected  at  A,  a  distance  beyond  the  red  of  twice  the  visible 
spectrum. 

Fig.   118  is  constructed  in  a  similar  way,  but  with  the 
sun  as  the  source  of  radiation.     The  light  part  is  the  visible 


FIG. 118. 

spectrum ;  the  dark,  that  beyond  the  red.  With  a  rock-salt 
prism  the  upper  curve  was  constructed,  with  a  flint-glass  prism 
the  lower  curve.  We  infer  that  flint  glass  stops  part  of  the 
dark  radiation.  (What  effect  would  you  expect  this  to  have  on 
the  temperature  of  the  flint-glass  prism  ?) 

Note,  in  the  case  of  sunlight,  that  the  dark  spectrum,  com- 
pared with  the  coloured  spectrum,  is  much  smaller  than  that 
obtained  from  the  electric  arc.  This  is  due  to  the  aqueous 
vapour  contained  in  the  atmosphere,  and  which  absorbs  the 
longer  waves  of  the  dark  radiation. 

The  meaning  of  the  letters  H,  G,  F,  etc.  (Fig.  118),  will 
be  given  later  (§  168). 

The  examination  of  the  invisible  spectrum  beyond  the  red 
(called  the  ultra-red  portion)  has  been  greatly  extended  by 
Langley,  who  has  invented  a  special  instrument  called  a 


Radiation  251 

bolometer  (Fig.  no).     By  its  aid  differences  of  of  a 

100000 

degree  Centigrade  can  be  detected. 

A  current  from  a  battery,  B,  passes  to  A,  and  there  divides. 
The  divided  currents  unite  again  at  C.  The  wires  carrying 
the  divided  currents  are  joined  through  a  delicate  galvano- 
meter, G.  a  and  b  are  two  delicate  coils  made  of  iron 

ribbons  —  of  an  inch  wide  and  -      -  of  an  inch  in  thick- 
50  12000 

ness.     When  the  resistances  of  a  and  b  are  equal,  and  both 
are  at  the  same  temperature,  no  current  passes  through  G. 


FIG.  119. 

A  delicate  resistance-box  (not  shown)  serves  to  make  the 
resistances  equal 

If  one  of  the  coils,  say  a,  be  heated,  its  resistance  is 
increased,  and  a  current  passes  through  G.  The  apparatus 
in  use  is  so  arranged  that  the  radiant  energy  passes  through 
a  narrow  slit  and  falls  upon  a,  the  heating  effect  being 
measured  by  the  deflection  of  the  galvanometer.  By  moving 
the  slit  along  the  spectrum,  the  heating  effect  both  of  the 
visible  and  the  ultra-red  portions  has  been  mapped  out. 

By  substituting  prisms  of  different  materials,  we  can,  by 
the  construction  of  curves,  come  to  some  general  conclusion 
as  to  the  effect  of  such  substances  in  absorbing  the  thermal 
rays.  If,  for  instance,  a  water  prism  with  sides  of  rock  salt  be 
substituted,  then  the  maximum  heating  effect  is  in  the  visible 
spectrum.  If  the  liquid  be  iodine  dissolved  in  carbon  di- 
sulphide,  the  visible  spectrum  disappears,  but  the  heating  effect 
can  still  be  observed. 

Bodies  that  transmit  thermal  rays  are  called  diathermanous , 
rock  salt  being  the  best  example;  the  term  adiathermanous 
being  used  for  substances  that  absorb  thermal  waves.  Both 


252  Heat 

terms  can  also  be  applied  to  waves  of  a  definite  frequency. 
The  portion  of  the  thermal  rays  absorbed  measures  the 
absorption  of  the  particular  medium. 

168.  Selective  Absorption. — If  the  strings  of  a  piano 
be  struck  in  turn,  each  produces  a  distinctive  fundamental 
note.  If  now  a  note  be  sung  or  sounded  near  these  strings, 
the  string  which,  when  struck,  produced  this  note  begins  to 
vibrate,  and  the  others  remain  unaffected.  Singing  the  note 
sets  the  particles  of  air  into  longitudinal  vibration,  and 
waves  of  definite  length  are  transmitted  by  the  air.  These 
waves  strike  all  the  strings,  but  only  a  particular  string  readily 
responds.  If  now  this  same  note  be  sung  into  a  large  number 
of  strings,  each  of  which  can  respond,  it  is  readily  seen  that 
the  energy  to  start  them  must  come  from  the  sound-wave, 
which  is  thereby  weakened,  and  we  can  even  imagine  a  con- 
dition in  which  it  will  be  altogether  extinguished. 

A  similar  selective  power  is  observable  in  radiant  energy. 
The  spectra  of  incandescent  solids  and  liquids  are  continuous ; 
that  is,  there  is  no  break  in  the  visible  spectrum  produced, 
and,  if  the  temperature  can  be  raised  high  enough,  the  whole 
of  the  visible  spectrum  appears,  save  that  the  intensity  of 
particular  colours  will  predominate  for  separate  substances. 
If  in  §  155  the  body  could  be  observed  through  a  prism  as 
its  temperature  rose,  red  would  appear  in  the  visible  spectrum 
first,  and  would  be  gradually  followed  by  the  other  colours, 
orange,  yellow,  green,  blue,  and  finally  by  violet. 

Instead  of  a  continuous  spectrum,  brightly  coloured  bands, 
with  dark  spaces  between,  are  obtained  from  vapours;  the 
physical  interpretation  being  that  ether-waves  of  certain 
definite  periods  only  are  started  by  the  glowing  vapours. 
Sodium  vapour,  for  example,  if  heated,  starts  waves  whose 
lengths  are,  in  air,  about  ^Sg-^^or  $89-9^,  their  effect  on  the 
eye  being  to  produce  the  sensation  of  yellow.  Such  rays,  after 
passing  through  a  prism,  appear  as  two  sharply  defined  yellow 
lines  close  to  each  other,  in  what  would  be  the  yellow  portion 
of  the  complete  visible  spectrum.  Now,  suppose  radiant 
energy  from  an  incandescent  solid  to  pass  through  sodium 
vapour ;  the  radiant  energy  is  being  transmitted  in  waves  of 


Radiation  253 

varying  lengths,  but  those  nearly  589* 3 /A/*  and  589-9/1/4  long 
lose  energy  in  starting  the  molecules  of  sodium  vapour  into 
motion ;  therefore  the  radiant  energy,  after  passing  through  the 
sodium  vapour,  is  deficient  in  these  particular  waves,  with  the 
result  that,  after  passing  through  a  prism,  spaces  or  dark  lines 
appear  in  the  spectrum  where  the  bright  yellow  lines  would 
otherwise  have  been.  On  this  account  the  solar  spectrum  is 
crossed  by  distinct  black  lines.  The  principal  of  these  were 
first  mapped  out  by  Frauenhofer,  and  lettered  A,  B,  C,  D, 
.  .  .  H(Fig.  1x8). 

At  such  dark  spaces,  seeing  that  either  no  radiant  energy 
or  a  smaller  quantity  than  before  is  received,  there  will  be  a 
reduced  heating  effect.  By  his  bolometer  Langley  has  shown 
that  dark  spaces  appear  also  in  the  ultra-red  spectrum.  From 
the  existence  of  dark  lines  in  the  solar  spectrum  we  infer  that 
the  radiant  energy  from  the  sun  has  passed  through  sodium  and 
other  vapours  in  its  photosphere,  and  we  can  come  therefore 
to  conclusions  concerning  the  composition  of  the  sun  and 
other  astronomical  bodies. 

As  illustrating  this  selective  absorption,  we  may  state  that 
lampblack  absorbs  rays  of  all  lengths :  the  absorption  is 
total.  Red  glass  absorbs  all  save  the  rays  that  produce  the 
sensation  of  red ;  blue  absorbs  all  save  those  that  produce 
the  sensation  of  blue ;  that  is,  each  kind  of  glass  selects  certain 
rays.  Rock  salt  is  peculiarly  diathermanous  to  the  shorter 
waves,  and  thus,  by  using  it  in  prisms,  very  little  heat  is  lost 
Alum  readily  allows  the  waves  that  produce  light  to  pass,  but 
absorbs  nearly  the  whole  of  the  ultra-red  waves.  If  iodine  be 
dissolved  in  carbon  disulphide,  a  solution  is  obtained  that  is 
opaque  to  light,  but  which  readily  transmits  the  heat-waves. 

The  relation  between  the  emissive  and  absorptive  powers 
of  bodies  for  radiant  energy  was  stated  by  Kirchhoff,  in  1860, 
in  this  form :  The  relation  between  the  powers  of  emission 
and  powers  of  absorption  for  rays  of  the  same  wave-length  is 
constant  for  all  bodies  at  the  same  temperature. 

In  the  preceding  sections  the  general  results  of  radiation, 
as  obtained  from  experiments,  have  been  given.  In  the 
following  sections  considerations  will  be  deduced  from  these 


254  Heat 

experimental  results.  It  will  be  impossible  to  obtain  in 
practice  all  the  conditions  that  will  be  imagined ;  for  perfect 
radiators,  perfect  absorbers,  perfect  reflectors  of  radiant 
heat,  the  existence  of  which  will  sometimes  be  assumed, 
are  not  at  our  command.  The  method  that  will  be  followed 
can  be  illustrated  from  mechanical  studies.  Questions 
relating  to  pulleys,  for  example,  are  discussed  by  imagining 
pulleys  that  work  without  friction,  and  cords  that  are  per- 
fectly flexible  (neither  condition  can  be  attained) ;  and 
the  law  deduced  is  applied  to  practical  cases  by  introducing 
the  effects  of  friction  and  rigidity.  The  subject  might  also 
have  been  treated  by  noting  the  relation  between  power  and 
weight  with  the  pulleys  and  ropes  at  our  command,  by  making 
the  pulleys  as  frictionless  as  possible,  and  by  using  cords  as 
flexible  as  possible,  by  noting  the  relation  between  power  and 
weight  under  these  conditions,  and  finally  determining  the 
relation  by  imagining  perfect  conditions. 

169.  Newton's  Law  of  Cooling. — The  following  con- 
ditions can  be  fulfilled :  In  a  room  without  any  fire  or  light, 
unexposed  to  the  direct  rays  of  the  sun,  the  temperature  of 
any  part  of  the  room  will  be  the  same,  and  will  keep  fairly 
constant.  This  is  nearly  attained  in  a  room  with  a  northern 
aspect  at  the  basement  of  a  building.  Near  the  middle  of 
such  a  room  place  a  vessel  containing  water  slightly  above  the 
temperature  of  the  room,  and  insert  vertically  in  the  vessel  a 
delicate  thermometer.  For  this  experiment  a  thin  brass  vessel 
that  will  contain  about  half  a  pint  of  water,  fitted  with  a  lid 
through  which  passes  a  delicate  thermometer,  is  suitable. 
Note  the  temperature  every  few  minutes,  at  first  frequently, 
and  try  to  determine  the  rate  at  which  the  water  cools. 
Suppose  the  temperature  of  the  room  is  16°  C,  and  the 
vessel  is  at  26°  C.,  that  is,  there  is  a  difference  of  10°  C. 
Note  the  number  of  seconds  it  takes  the  temperature  of  the 
vessel  to  fall  one  degree,  that  is,  from  26°  to  25°,  from  25°  to 
24°,  etc.  In  falling  from  26°  to  25°,  the  difference  in  tempera- 
ture between  the  room  and  the  vessel  at  the  beginning  is  10°, 
at  the  end  9°.  Let  us  take  9^°,  or  19  half-degrees,  as  the  mean 
difference ;  some  result  like  the  following  will  be  obtained  : — 


Radiation 


255 


Fall  in  temperature  of 
vessel. 


Mean  difference 

in  temperature  between 

room  and  vessel,  in 

half-degrees. 


Time,  in 
seconds. 


Fn 

)m  26°  t 

025°  ... 

19 

60 

25° 

24°  ... 

17 

67 

24° 

23°  ... 

15 

76 

23° 

22°    ... 

13 

88 

22° 

21°    ... 

ii 

104 

21° 

2O°    ... 

9 

127 

20° 

19°    ... 

7 

'£ 

18°  ... 

5 

228 

1  8° 

17°  ... 

3 

380 

Experiments  can  be  made  in  any  room  shaded  from  the  sun. 

The  general  fact  that  the  greater  the  difference  in  tempera- 
ture the  quicker  the  rate  of  cooling,  is  obvious  ;  also  that 
when  the  difference  is  5  half-seconds,  the  rate  is  one-third 
of  the  value  it  has  when  the  difference  is  15  seconds.  If  we 
examine  the  relation  between  the  rate  of  cooling  and  the 
differences  in  temperature,  we  obtain  the  following  table  : — 


Difference  in 
temperature,  in 
half-degrees. 
t 

Rate  of  cooling, 
in  degrees 
per  second. 

V 

Ratio. 
v         ^ 

7 

19 

A 

edbs  T=  n1  TO 

17 

eV 

8T^TT=Tn!9 

15 

7!S 

etc.  =  TTW 

13 

* 

=  n'« 

II 

ife 

=  TTI« 

9 

1*7 

=11^ 

7 

ib        - 

=iAi 

5 

db 

=  n'TO 

3 

:li, 

=  r/* 

For  the  series  of  experiments  we  see  that   v-    practically 

equals  a  constant  (K)  (average  value  nearly  ~^~\  •  that  is— 

11417 

v  =  K/ 
This  expresses  Newton's  law  of  cooling— that  the  rate  of 


256  Heat 

cooling  varies  directly  as  the  excess  of  the  temperature  of  the 
body  over  that  of  the  enclosure. 

The  rate  of  change  of  temperature  of  a  vessel  varies 
directly  as  the  difference  between  the  temperature  of  the 
vessel  and  the  enclosure. 

This  "law  "  is  fairly  correct  when  the  difference  of  tempera- 
ture is  small ;  it  does  not  apply  when  the  difference  of  tempera- 
ture is  great. 

v  =  K/ 

The  constant,  K,  will  vary  with  the  surface  of  the  vessel, 
its  area,  the  substance  it  contains,  and  the  mass  of  the  sub- 
stance. 

A  series  of  experiments  with  the  vessel  slightly  below  the 
temperature  of  the  room,  say  at  10°  C.,  would  show  that  the  rate 
of  heating  of  the  vessel  varied  in  a  similar  manner. 

170.  Dulong  and  Petit's  Law  of  Cooling.— The 
essential  parts  of  the  apparatus  of  these  two  experimenters 
consisted  of  a  small  hollow  brass  sphere,  into  which  fitted 
vertically  a  small  metal  cylinder.  Through  this  could  be 
passed  and  fixed  an  ordinary  thermometer  openly  divided ;  a 
closed  glass  tube  slipped  over  the  thermometer  when  in 
position.  The  inside  of  the  sphere  was  blackened  with  lamp- 
black ;  the  interior  was  also  connected  with  an  air-pump. 
The  sphere  was  placed  in  a  vessel  of  water  so  that  it  was 
completely  covered ;  the  water  could  be  kept  at  any  required 
temperature  (for  temperatures  above  100°,  oil  could  be  used), 
being  heated  when  necessary  by  passing  steam  into  it.  The 
brass  was  thin,  therefore  the  temperature  of  the  interior  surface 
was  that  of  the  water,  6°. 

The  thermometer  was  taken  out  and  heated  until  its 
temperature  was  about  350°  C.,  then  replaced  so  that  its  bulb 
was  at  the  centre  of  the  sphere,  and  the  sphere  was  exhausted 
as  far  as  practicable.  Thus  the  conditions  were  a  small  body 
(the  bulb  of  the  thermometer)  at  a  temperature,  0  +  t  (t  being 
the  difference  between  the  temperature  of  the  enclosure  and 
that  of  the  thermometer),  surrounded  by  an  enclosure  whose 
temperature  was  0,  the  air  between  the  two  being  exhausted 


Radiation 


257 


Observations  were  made  with  the  temperature  of  the 
water  (and  therefore  that  of  the  interior  of  the  sphere)  at  o°, 
20°,  40°,  etc.  (called  6  in  the  table  below),  and  the  relation 
between  the  rate  of  cooling  and  difference  of  temperature  was 
calculated,  the  object  being  to  ascertain  whether  Newton's  law 
was  independent  of  the  absolute  temperatures  of  the  vessels. 


Differences 
in  degrees. 
t 

Rate  of  cooling  for  various  values  of  9. 

,=  0. 

0  =  20. 

0  =  40. 

,  =  60. 

240 
2OO 
100 
IO 

7-40 
2-30 

I2-40 

2-74 
I-40 

lO'OI 

1-62 

II-64 

3'68 
1-88 

Examining  any  horizontal  line  in  the  table,  we  see  that  the 
rate  of  cooling  increases  with  the  temperature  of  the  enclosure, 
although  a  constant  difference  of  temperature  is  maintained 
(see  Table). 


I  2  '40  _ 


=  ri6j 


°  =  rx6; 

7-40 


=  ri6 


12*40 

TO'OI 


8-58  10-01 

Proceed  similarly  with  the  other  numbers  to  verify  that  the 
quotient  is  practically  constant,  and  equal  to  1*16  =  c. 

Suppose  we  have  found  the  absolute  rate  of  cooling  for  any 
one  of  the  differences.  When  the  enclosure  is  at  o°,  call  this 
value  /x  •  then  we  can  write  any  line  as  follows  : — 


Difference. 
t 

•— 

0  =  20. 

0  =  ^0. 

0=60. 

Or 

J* 

£ 

$> 

£ 

Or 

Or,  calling 
c  &  =  a 

*£f 

>" 

**? 

>f 

rate  of  cooling  v  = 


258  Heat 

where  we  have  to  determine  /*,  so  that  it  will  apply  to  any 
line. 

Dulong  and  Petit  made  the  assumption  that  the  rate  of 
cooling  depended  in  some  way  upon  the  temperature  of  the 
thermometer  (t  +  0)  and  upon  the  temperature  of  the  enclosure 
(0),  and  calculated  that  the  velocity  of  cooling  under  any  con- 
dition could  be  represented  by — 

v  =  mae(at  —  i) 

where  6  is  the  temperature  of  the  enclosure ;  /,  the  difference 
of  temperature  between  enclosure  and  thermometer,  m  is  a 
constant  in  any  series  of  experiments,  and  depends  upon  the 
nature  of  the  surface,  the  masses,  and  the  kind  of  matter. 

a20  =  <r  =  1-16 

20 /— — 7 

.-.  a  -  <\/i'i6  =  1*0077 

temperature  being  measured  on  the  Centigrade  scale. 

In  arriving  at  this  formula  assumptions  were  made;  its 
value,  therefore,  depends  upon  the  question  whether  values 
calculated  from  the  formulae  agree  with  the  facts  as  obtained 
from  experiments. 

Suppose  the  temperature  (0)  of  an  enclosure  remains  constant 
in  a  series  of  experiments,  then — 

v  =  ma9  (a  —  i)  =  K(a  —  i) 
where  K  is  a  constant  equal  to  m(\'o^Y. 
v  =  K(ro77   -  i) 

Giving /the  values  i,  2,  3,  etc.,  we  obtain  a  table  comparing 
v  as  determined  by  Dulong  and  Petit's  formulae,  and  as 
determined  by  Newton's  (v  =  K/),  assuming  that  the  two  laws 
give  the  same  result  when  the  difference  is  i°. 


Radiation 


259 


Rate  of  cooling. 

Difference  of 
temperature. 

Dulong  and 
Petit. 

Newton. 

_           \Tf 

Z/=K(>*-l) 

1° 

0-0077 

0-0077 

2° 

0-0155 

0-OI54 

3° 

0-0233 

0-023I 

5° 

0-039I 

0-0385 

10° 
20° 

0-0797 
0-1658 

0-0770 
0-I540 

(All  the  numbers  in  the  second  and  third  columns  should  be 
multiplied  by  K.) 

The  difference  is  slight  for  small  differences  of  temperature. 
Newton's  result  gives  a  value  about  3^  per  cent,  too  low  when 
the  difference  is  10°;  when  it  is  20°  the  value  of  the  difference 
is  7  per  cent.  The  error  is  so  small  for  small  differences 
that  we  may  accept  and  use  the  simple  law  of  Newton,  but 
when  the  difference  is  more  than  10°  the  divergence  is  too 
great  to  be  neglected. 

Other  researches  of  Provostaye  and  Desains  further  verify 
Dulong  and  Petit's  law  within  certain  limits. 

The  student  will  remember  that  Newton's  and  Dulong  and 
Petit's  laws  apply  to  the  total  radiation.  The  existence  of 
air  will  produce  air-currents,  and  heat  will  be  transferred  by 
convection. 

171.  Emissivity  and  Absorption. — "  Thermal  emissivity 
is  the  quantity  of  heat  per  unit  of  time,  per  unit  of  surface,  per 
degree  of  excess  of  temperature,  which  the  isolated  body  loses 
in  virtue  of  the  combined  effect  of  radiation  and  convection  by 
currents  of  air." 

Therefore,  the  loss  in  time,  /,  will  be — 

Q  =  ES0/ 

S  =  area  of  surface,  6  =  difference  in  temperature  between  the 
body  and  the  enclosure,  t  -  time,  and  E  a  constant  for  the 
particular  substance.  E  is  the  coefficient  of  emissivity,  or 


260  Heat 

E  will  be  the  coefficient  of  absorption,  if  the  body  be  below 
the  temperature  of  the  enclosure. 

Macfarlane  obtained  the  emissivity,  in  C.G.S.  units,  of 
copper,  blackened  and  bright,  in  air,  at  ordinary  pressure  ;  the 
walls  of  the  enclosure  were  blackened,  and  were  at  14°  C. 

Differences  of  Emissivity. 

temperature.  Polished  surface.  Blackened  surface. 

10°     ^j..      0*000186        ...       0*000266 

...          0-OO0279 


...  O*OOO20I 

° 


50°          ...  0*000225  ...  O'OOO326 

The  meaning  of  one  of  the  numbers,  say  0*000356,  is  that 
when  blackened  copper  at  64°  C.  is  surrounded  with  damp  air 
at  atmospheric  pressure  enclosed  in  a  vessel  whose  temperature 
is  14°,  then  the  total  loss  of  heat  due  to  radiation  and  con- 
vection from  i  square  cm.  of  the  blackened  surface  is  0*000326 

thermal  unit  (—  *—  nearly^ 
^3070  ) 

Professor  Tait  gives  the  following  numbers  obtained  by 
Nicol  :  — 

Hot  body,  58°  C.  ;  enclosure,  8°C.  ;  pressure  of  contained 
air  in  inches  of  mercury,  (a)  30,  \b)  4,  (c)  0*4.  Units  :  i  lb., 
i  foot,  i  minute,  i°  C.,  and  the  thermal  unit. 

(«)  (*)  (c) 

Bright  copper        ...        1*09       ...       0*51       ...       0*42 
Blackened  „  ...       2-03       ...       1-46       ...       1-35 

The  numbers  represent  the  total  loss. 

To  compare  these  numbers  with  the  above,  we  must  change 
the  units. 

The  dimensions  of  emissivity  are  obtained  from  Q  -4-  S$/. 
The  dimensions  of  Q  =  MA,  S  =  L2,  0  =  A,  /  =  T  (§  75). 

MA         M 
/.  the  dimensions  of  emissivity  E  =  •\2\rr  =  TjT 

.'.  for  total  loss  for  0°  dimensions  of  total  emissivity  =    - 


Radiation  261 

(453-6  grams)(C°)          .  (gram)(C.°) 


(foot)2  (minutes)      (30-48  cm.)2  (60  sees.)  (cm.)  (sees.) 

.'.  the  multiplier  is  0-00805 

Therefore  from  the  last  table  we  obtain  at  a  pressure  in 
millimetres  of  mercury  :  (a)  760,  (b)  ioi'6,  (c)  io'2. 

GO  (*)  (c) 

Bright  copper     ...     0-00887     ...     000415     ...     0*00342 
Blackened   „      ...     o-oi652     ...     001188     ...     0-01099 

Dividing  the  first  column  by  50,  we  obtain  the  average  emis- 
sivity  per  difference  of  i  C.°  :  0-000177  (bright),  and  0-000330 
(blackened),  numbers  that  agree  fairly  with  Macfarlane's. 

172.  Prevost's  Theory  of  Exchanges.  —  The  theory  is 
that  all  bodies  are  radiating  heat,  whatever  their  temperature  may 
be,  and  that  the  radiation  depends  upon  the  temperature  of 
the  body  and  the  body  itself,  and  not  upon  the  temperature 
of  surrounding  objects.  It  follows  that  a  red-hot  ball  radiates 
the  same  quantity  of  heat,  whether  in  an  ice-chamber  or  in  the 
middle  of  a  furnace,  provided  its  temperature  is  the  same  in 
both  experiences.  Applied  to  a  body  in  an  enclosure,  we 
have  the  result  —  the  radiation  of  the  body  is  independent  ot 
the  temperature  of  enclosure.  The  enclosure  is  also  radiating 
heat  to  the  body;  if  the  temperature  of  the  body  rise,  it  is 
receiving  more  radiant  heat  than  it  loses  ;  if  its  temperature 
falls,  it  is  radiating  more  than  it  receives  ;  if  the  temperature 
remains  constant,  then  there  is  a  balance  between  the  amount 
radiated  and  the  amount  absorbed.  A  warm  body  not  only 
radiates  heat  to  a  mass  of  ice,  but  the  ice  radiates  heat  to  the 
warm  body  (see  §  159). 

From  highly  polished  metal  surfaces  the  radiation,  as  we 
have  seen,  is  very  small.  If  we  construct  an  enclosure  with 
a  bright  metal  exterior,  then  it  radiates  a  very  small  amount  ; 
also,  since  it  reflects  all  radiant  heat,  it  absorbs  little  ;  if  we 
further  place  a  hot  or  cold  body  in  such  an  enclosure  at  any 
temperature,  then  after  a  certain  time  the  body  and  enclosure 
will  be  at  the  same  temperature,  and  will  both  keep  at  that 
temperature.  This  is  a  fact  that  we  deduce  from  experiments, 
when  we  eliminate  as  far  as  possible  the  gain  or  loss  of  heat 


262  Heat 

from  external  objects.  The  body  is  radiating  heat  to  the 
enclosure,  and  is  absorbing  heat  from  the  enclosure  ;  and  as  its 
temperature  remains  constant,  its  radiating  power  must  equal 
its  absorbing  power,  whatever  the  nature  of  the  body  may  be 
(compare  the  construction  of  a  good  calorimeter).  This  is 
dealing  with  the  total  radiation.  It  is  also  true  of  any  par- 
ticular part  of  the  radiation.  Stewart  illustrated  this  by 
examples  similar  to  the  following : — A  piece  of  blue  glass 
ordinarily  appears  blue  because  it  absorbs  the  longer  light-waves 
(i.e.  the  red)  and  transmits  the  blue.  If  such  a  piece  of  glass 
be  placed  in  the  middle  of  a  fire,  its  blue  colour  seems  to 
disappear,  and  it  cannot  be  distinguished  from  the  surrounding 
hot  portion  of  the  fire.  It  still  transmits  the  blue  rays,  but, 
being  heated  to  a  high  temperature,  it  radiates  also  red  waves ; 
these  two  together  produce  the  sensation  of  reddish  white. 
To  further  test  this  theory,  if  the  hot  glass  be  taken  into  a 
dark  room,  it  no  longer  appears  blue,  but  reddish,  i.e.  it  is 
radiating  the  waves  it  previously  absorbed. 

173.  Crookes's  Radiometer.— A  glass  tube  is  blown  into 
a  globular  shape  at  its  upper  part  (Fig.  120).  Into  the  lower 
part  is  fused  a  thin  upright  glass  -tube,  that,  fitted  into  a  wooden 
stand,  supports  the  whole  of  the  apparatus.  A  fine  steel  point 
is  fused  into  the  upper  part  of  this  tube,  and  on  the  point 
rests  a  small  glass  or  agate  cap.  To  the  cap  are  attached 
four  aluminium  wire  arms  in  a  horizontal  plane,  making  a 
right  angle  with  each  other ;  the  arms  carry  small  diamond- 
shaped  discs  of  mica,  each  in  a  vertical  plane.  A  glass  tube 
is  fused  into  the  upper  portion  of  the  apparatus,  its  purpose 
being  to  keep  the  cap  in  position  when  the  whole  is  moved 
about ;  the  cap,  however,  does  not  touch  this  tube  when  the 
discs  are  moving.  The  upper  end  of  this  tube  is  open,  and, 
by  connecting  it  with  a  Sprengel  pump,  the  air  inside  the 
instrument  can  be  exhausted;  the  opening  is  then  sealed. 
The  aluminium  discs  are  covered  on  one  side  with  lampblack, 
the  other  side  remaining  bright,  and  they  are  so  arranged  that 
a  black  side  is  always  followed  by  a  bright  side  as  we  move 
round  the  four  discs. 

If  the  instrument,  as  ordinarily  sold,  be  exposed   to  a 


Radiation 


263 


source  of  radiant  energy,  the  discs  move  in  such  a  way  that 
the  black  sides  move  away  from  the  source  of  the  radiant 
energy,  which  may  be  a  lamp, 
the  sun,  etc. ;  the  greater  the 
radiant  energy,  the  quicker  the 
rate  of  revolution.  Thus,  by 
considering  the  number  of  revo- 
lutions and  the  distance  of  the 
source,  the  intensity  of  the 
radiation  may  be  measured. 

If  the  opening  at  the  top 
be  kept  open,  and  connected 
with  a  pump,  so  that  the  amount 
of  exhaustion  can  be  regulated 
at  will,  then  for  any  source  of 
radiant  heat  it  is  found  that, 
after  a  certain  degree  of  exhaus- 
tion, the  following  results  are 
observable :  The  black  discs 
are  repelled,  and  rotation  en- 
sues. If  the  exhaustion  be 
gradually  increased,  a  maximum 
speed  is  reached ;  further  ex- 
haustion diminishes  the  number 
of  revolutions,  and  ultimately 
the  rotation  ceases. 

Crookes  showed  that  the 
best  effects  were  obtained  if  the 
globe  were  as  small  as  possible. 
It  has  also  been  shown  that,  if 
a  specially  constructed  instru- 
ment of  great  delicacy  be  made 
with  brass  discs,  and  floated  in  water,  and  if  the  discs  be  kept 
in  position  by  a  strong  magnet,  then,  when  radiant  energy 
acts  upon  the  discs,  the  globe  rotates  in  the  water.  This 
shows  that  the  action  and  reaction  must  take  place  between 
the  disc  and  the  glass  globe,  and  not,  as  it  might  appear, 
between  the  discs  and  the  source  of  light  or  heat. 


FIG.  120. 


264  Heat 

The  experimental  results  show  that,  in  the  working  instru- 
ments, there  is  not,  as  was  at  first  supposed,  a  perfect  vacuum, 
but  that  there  is  in  all  cases  a  residue  of  gas  or  air  left. 

The  molecules  of  gas  or  air  are  moving  about  in  all  direc- 
tions, with  a  speed  that  depends  upon  the  temperature  (§  77). 
If  the  degree  of  exhaustion  be  sufficient,  the  molecules  possess 
a  free  mean  path  equal  to  or  greater  than  the  distance  between 
the  disc  and  the  surface  of  the  glass ;  that  is,  a  molecule,  after 
impinging  upon  the  discs,  does  not  collide,  as  a  rule,  with 
any  other  molecule  before  reaching  the  glass. 

When  subjected  to  any  source  of  radiant  energy,  the  black 
side  absorbs  more  heat  than  the  metal  side ;  the  temperature 
of  the  black  side  is  therefore  greater  than  that  of  the  metal 
side. 

Let  us  suppose  that  at  the  beginning  a  black  side  of  one 
disc  and  the  metal  side  of  a  disc  two  right  angles  away  are 
directed  towards  the  source  of  heat  or  light,  we  can  then 
neglect  the  heating  effect  on  the  other  two  discs.  The  mole- 
cules at  first  are  impinging  on  all  sides  of  all  the  vanes  with 
equal  speeds,  and  are  rebounding  with  the  same  speeds. 
When  the  source  of  light  or  heat  is  brought  near,  the  tempera- 
ture of  the  black  side  rises  above  that  of  the  metal  side ;  there- 
fore the  molecules  will  be  heated  and  will  rebound  with 
greater  speed  from  the  black  side ;  this  recoil  is  thus  greater 
than  the  recoil  from  the  metal  side,  and,  by  the  third  law  of 
motion,  the  disc,  being  free  to  move,  will  move  so  that  the 
black  side  moves  away  from  the  source  of  light  or  heat.  The 
action  continues  to  a  less  degree  when  its  surface  makes  an 
angle  with  the  direction  in  which  the  radiant  energy  moves ; 
another  disc  is  brought  into  position,  and  a  continuous  rotation 
ensues.  If  the  radiometer  be  now  further  exhausted,  there  are 
fewer  molecules;  the  difference  between  the  effects  of  the 
bombardment  on  the  two  sides  will  be  lessened,  and  the  speed 
of  rotation  will  be  diminished.  Ultimately  the  difference  will 
be  insufficient  to  overcome  the  slight  friction,  and  rotation 
ceases. 

174.  Solar  Radiation. — To  determine  the  solar  radia- 
tion, Pouillet  constructed  an  instrument  called  a  pyrheliometer. 


Radiation  265 

It  consists  of  a  cylindrical  vessel  of  thin  metal,  highly  polished 
(Fig.  121).  The  upper  surface  is  covered  with  lampblack. 
The  bulb  of  a  thermometer  is  inserted  in  the  cylinder,  which 
is  otherwise  filled  with  water;  the  stem  of  the  thermometer 
is  further  enclosed  in  another  tube.  The  lower  disc  is  of  the 
same  diameter  as  the  cylinder; 
its  use  is  to  ensure  that  the 
blackened  face  is  turned  di- 
rectly towards  the  sun.  When 
this  is  the  case,  the  shadows 
of  the  cylinder  and  the  disc 
will  coincide. 

The  instrument  at  the  be- 
ginning of  an  observation  is 
shaded,  and  is  at  the  tempera- 
ture of  the  air.  The  lamp- 
black face,  carefully  shaded 
from  the  sun,  is  now  turned 
towards  the  sky  for  a  definite 
time  (/  units)  ;  radiation  takes 
place,  and  the  temperature 
falls  6  degrees. 

The  face  is  now  turned 
towards  the  sun  for  the  same 
time  ;  the  temperature  rises  ® 
degrees.  It  is  next  turned 

towards  the  sky,  as  at  the  first,  for  the  same  time,  and  the 
temperature  falls  &  degrees. 

During  the  time  it  was  turned  towards  the  sun  it  absorbed 
and  also  radiated  heat.  The  fall  in  the  temperature  due  to 
the  radiation  for  the  time  when  its  initial  temperature  is  that 
of  the  air  is  0  degrees;  when  the  initial  t  temperature  is  the 
highest  point  registered,  it  is  6'.  It  is  assumed  that  the  total 
radiation  during  the  exposure  to  the  sun  is  the  mean  of  these 
two ;  that  is — 


266  Heat 

Therefore  the  total  rise  of  temperature  that  would  have 
been  registered  if  there  had  been  no  radiation  would  have 
been  — 


When  radiant  energy  from  the  sun  falls  upon  an  area  equal 
to   that  of  the   lampblack   face   (A),  it  is  able  to  raise  the 

fi  -L-  fi1 

temperature    of  the  vessel  ®  -f  —  -  degrees.      The   water 


equivalent  of  the  vessel,  the  water,  and  the  thermometer  bulb 
can  be  calculated.  Let  it  be  <o.  Then  the  radiant  energy 
received  per  unit  area  per  unit  time  is  — 


Radiant  energy  is  absorbed  by  the  atmosphere,  and  Pouillet 
attempted  to  determine  the  effect  of  the  thickness  of  the 
atmosphere  upon  the  heat  we  receive  from  the  sun,  and 
concluded  that  not  more  than  one-half  of  the  total  radiation 
reached  the  earth. 

We  are  able  to  measure  the  unabsorbed  radiation  that 
reaches  us  in  units  of  heat  per  unit  area  per  unit  of  time,  and, 
by  using  Joule's  equivalent,  are  able  to  calculate  the  energy  in 
mechanical  units. 

If  the  whole  were  employed  in  melting  ice,  it  would  in 
a  year  melt  a  layer  of  ice  all  round  the  earth  35  yards  in 
thickness.  Other  calculations  state  the  unabsorbed  energy  as 
equal  to  133  foot-pounds  per  square  foot  per  second. 

Only  a  fraction  of  the  sun's  heat  reaches  the  earth.  Sir 
William  Thomson's  estimate  is  that  the  total  energy  of  the  sun 
is  emitted  at  the  rate  of  7000  horse-power  per  square  foot. 

EXAMPLES.    XI. 

1.  Explain  the  terms  "wave,"  "wave-length,"  "frequency,"  "ampli- 
tude." 

2.  The  sun  radiates  his  heat  towards  the  earth,  and  the  earth  radiates 
her  heat  through  stellar  space  :  do  you  suppose  that  the  solar  and  terres- 
trial heat  passes  with  the  same  ease  through  the  earth's  atmosphere  ?    Give 
the  effect  of  your  answer  on  the  temperature  of  the  earth. 


Radiation  267 

3.  Give  an  example  of  a  good  and  a  bad  radiator  ;  also  of  a  good  and 
a  bad  absorber  ;  and  state  the  general  relation  that  exists  between  radiation 
and  absorption. 

4.  If  two  planets  of  equal  size  be  placed,  one  at  100,  and  the  other  at 
200  millions  of  miles  from  the  sun,  compare  the  heat  received  by  each  per  unit 
of  surface.     Imagine  and  describe  an  atmosphere  by  which,  if  the  distant 
planet  be  surrounded,  its  temperature  will  become  equal  to  or  even  greater 
than  the  near  one. 

5.  A  very  sensitive  thermometer  has  its  bulb  covered  with  lampblack 
in  ordinary  air ;  a  current  of  perfectly  dry  air,  of  precisely  the  same  tem- 
perature as  the  surrounding  atmosphere,  is  urged  very  gently  against  the 
bulb ;  the  thermometric  column  sinks :  why  ?    On  stopping  the  current 
of  dry  air,  the  column  instantly  rises  :  why  ? 

6.  Explain  how  to  determine  the  emissive  powers  and  the  absorbing 
powers  of  substances  for  radiant  heat,  and  state  the  relations  which  exist 
between  them. 

7.  Explain  the  terms  "radiant  light"  and  "radiant  heat."    What  ob- 
jections can  you  raise  to  the  terms  ? 

8    Explain  why  the  temperature  of  a  greenhouse  is  above  that  of  the 
surrounding  air.     (The  greenhouse  is  unheated.) 

9.  How  could  you  separate  the  dark  radiation  of  an  arc  light  from  the 
luminous  radiation  ?  and  how  could  you  demonstrate  the  existence  of  the 
longer  rays  ? 

10.  Compare    Newton's  with   Dulong  and    Petit's  law  of   cooling. 
How  would  you  experimentally  demonstrate  the  laws  ? 

n.  Describe  and  explain  the  action  of  Crookes's  radiometer. 

12.  Illustrate  the  term  "selective  absorption." 

13.  A  thermometer  placed  in  an  open  black  box,  and  exposed  to  the  sun, 
rises  to  80°  F. ;  a  glass  cover  is  placed  on  the  box,  and  the  temperature 
rises  to  120°  F.  :    explain  this,  and  apply  your  explanation  to  the  possible 
influence  of  a  planet's  atmosphere  in  augmenting  its  temperature. 

14.  Explain  Prevost's  theory  of  exchanges,  and  apply  it  to  the  apparent 
"radiation  of  cold,"  in  §  159. 

15.  The  total  emissivity  of  glass  at  100°  C.  per  square  foot  per  minute 
to  an  enclosure  at  o°  C.,  is  0*176  thermal  unit  (unit  of  mass  is  i  lb.). 
Change  this  into  C.G.S.  units. 


268  Heat 


CHAPTER  XII. 

THE  RMO-D  YNA  MICS. 

175.  First  Law  of  Thermo-dynamics. — The  mechani- 
cal equivalent  of  heat  has  already  been  determined  from  Joule's 
experiments  to  be  772  foot-pounds  when  degrees  are  measured 
on  the  Fahrenheit  scale,  and  1390  foot-pounds  when  the  Centi- 
grade scale  is  used   (§  76),  and  from  such  experiments  the 
first  law  of  thermo-dynamics  is  deduced — 

"When  equal  quantities  of  mechanical  effect  are  produced 
by  any  means  whatever  from  purely  thermal  sources,  or  lost  in 
purely  thermal  effects,  equal  quantities  of  heat  are  put  out  of 
existence,  or  are  generated." 

H  (in  thermal  units)  =  JW  (in  mechanical  units) 

176.  Mayer's   Experiments. — Mayer    determined    the 
mechanical  equivalent  of  heat  from  observations   on   gases. 
His  method  is  illustrated  in  the  next  paragraph. 

The  following  experiment  is  supposed  to  be  performed 
when  the  atmospheric  pressure  is  normal,  that  is,  147  Ibs. 
on  the  square  inch.  Imagine  a  long  hollow  cylinder  placed 
vertically,  closed  at  the  bottom;  the  internal  section  is  i 
square  foot.  A  weightless  piston  fits  exactly  into  the  cylinder 
and  slides  in  it  without  friction.  This  piston  cuts  off  i  cubic 
foot  of  dry  air  at  o°  C.  and  147  Ibs.  pressure;  that  is,  the 
piston  is  i  foot  from  the  bottom  of  the  cylinder. 

The  area  of  the  piston  is  144  square  inches;  therefore  the 
total  pressure  on  the  piston  is  147  x  144  =  2116*8  Ibs. 

Suppose  heat  to  be  supplied  to  the  imprisoned  cubic  foot  of 
air  sufficient  to  double  its  volume. 


Thermo-  Dynamics  269 

The  coefficient  of  expansion  being  -  —  ,  we  know  that  the 

temperature  will  be  273°  C.  when  the  volume  is  doubled. 

The  external  work  is  that  done  in  overcoming  a  pressure  of 
2116-8  Ibs.  through  i  foot,  that  is,  2116  -8  foot-pounds  of  work. 

The  thermal  units  needed  to  raise  the  temperature  of  the 
confined  air  from  o°  to  273°  C.  and  to  do  the  external  work 
will  equal  the  product  of  mass,  specific  heat  at  constant 
pressure,  and  273. 

The  mass  of  i  cubic  foot  of  air  at  o°  C.  and  at  atmospheric 
pressure  is  o'oSi  Ib.  The  specific  heat  of  air  at  constant 
pressure  is  0*237. 

/.  the  thermal  units  used  in  (i)  raising  the  temperature,  that 
is,  in  increasing  the  kinetic  energy  of  the  particles,  and  (2) 
doing  external  work  =  o'oSi  x  0-237  *  273  =  5'24T  tner~ 
mal  units 

/.   5  -241  thermal  units  have  done  work  — 

(a)  Internally,  by  raising  the  temperature  ;  and 

(b)  Externally,  against  the  atmosphere. 

Now,  imagine  the  cubic  foot  of  air  again  at  o°  C.  and  at 
normal  atmospheric  pressure,  but  enclosed  in  a  rigid  vessel,  so 
that  expansion,  and  therefore  external  work,  is  impossible. 
Under  such  conditions  the  specific  heat  of  air  (at  constant 
volume)  is  0-168. 

/.  the  thermal  units  required  to  raise  the  temperature  from  o° 
to  273°  C.  at  constant  volume  =  0-081  x  o'i68  X  273 
=  3715  thermal  units 

Th.  units  Internal  work.  External  work. 


/  raising  the  temperature  ) 
-715  M       from  Q°  to  273°  C.      I 


.  1-526  =•  o  +  2116-8 

2116-8  p 
:.  i  thermal  unit  =        g6  foot-pounds 

=    1387 


270  Heat 

Mayer's  first  published  results  were  inaccurate,  but  this  arose 
from  the  errors  in  the  data  he  took  from  the  experimentalists. 
With  the  use  of  later  values  of  the  specific  heats  of  air,  results 
are  obtained  that  fairly  approximate  towards  Joule's  number. 

There  is  an  assumption  in  the  reasoning  that  prevents  the 
conclusion  being  a  logical  one.  It  is  assumed  that  the  energy 
necessary  to  raise  the  temperature  from  o°  to  273°  C.  without 
increase  in  volume,  is  equal  to  the  energy  required  to  raise 
the  temperature  from  o°  to  273°  C.  when  the  volume  is  doubled, 
provided  that  in  expansion  the  air  does  not  overcome  external 
resistance.  We  can  write,  for  example,  the  first  line  thus — 

5*241  thermal  units  =  (a)  heat  used  in  raising  the  temperature 
of  i  cubic  foot  of  air  from  o°  to  273°  C.,  the  volume 
remaining  constant  -f  (b)  internal  work  done  by  or  on 
the  air  in  expanding  from  i  cubic  foot  to  2  cubic  feet  + 
(c)  external  work  in  overcoming  resistance  equal  to  2116*8 
foot-pounds 

Now,  Mayer  assumed  that  (b),  the  internal  work  done  by  or 
on  the  air  in  expanding,  was  nil.  This  is  nearly  true  in  the 
case  of  air ;  it  is  exactly  true  in  the  case  of  a  perfect  gas ;  but 
it  is  far  from  true  in  the  case  of  many  other  gases. 

If,  for  example,  Mayer  had  'based  his  reasoning  on  the 
behaviour  of  carbon  dioxide  under  ordinary  conditions,  or  on 
the  behaviour  of  steam,  his  reasoning  would  have  been  yet  less 
reliable.  There  is  no  reason  given  for  the  selection  of  air, 
and  therefore  his  line  of  argument  is  not  logical. 

If,  however,  (b)  were  proved,  then  the  method  followed  by 
Mayer  can  be  used  for  determining  the  mechanical  equiva- 
lent of  heat  from  gases.  The  experiments  on  the  subject  are 
due  to  Joule  and  Thomson. 

177.  Joule's  Experiments  on  the  Internal  Work  of 
Expanding  Gases. — Two  strong  vessels  were  connected  by 
a  narrow  tube  in  which  was  a  stop-cock.  The  stop-cock  being 
closed,  air  was  forced  into  one  vessel  until  the  pressure  was 
22  atmospheres;  the  air  was  exhausted  from  the  other,  and  as 
perfect  a  vacuum  as  possible  produced. 

Both  were  placed  in  a  vessel  of  water  that  formed  a  calori- 


Thermo- Dynamics  271 

meter,  so  that  both,  together  with  the  joining  tubes,  were 
entirely  immersed.  The  water  was  kept  thoroughly  stirred, 
and  the  temperature  read  from  delicate  thermometers.  When 
the  temperature  was  constant,  the  stop-cock  was  turned; 
the  air  rushed  into  the  vacuum  and  expanded  to  twice  its 
volume,  and,  as  it  overcame  no  resistance,  it  did  no  external 
work.  No  change  could  be  detected  in  the  thermometers,  and 
Joule  concluded  that,  when  air  expands  without  doing  external 
work,  there  is,  on  the  whole,  neither  gain  nor  loss  of  heat; 
that  is,  that  there  is  no  internal  work  done  in  expanding  the 
gas,  otherwise  the  energy  necessary  must  have  been  taken 
from  the  heat-energy  of  the  gas,  with  a  consequent  fall  in 
temperature. 

There  might,  however,  be  a  loss  or  gain  of  heat  in  one 
vessel,  and  an  equal  gain  or  loss  of  heat  in  the  other,  and 
to  test  this  the  experiment  was  modified.  Each  vessel  was 
placed  in  a  separate  calorimeter.  In  one  vessel  (A)  the  air 
was  again  compressed ;  in  the  other  (B)  the  vacuum  was 
produced.  The  water  in  each  calorimeter  was  well  stirred, 
and  the  temperature  noted.  On  turning  the  stop-cock  the  air 
rushed  from  A  to  B,  and  a  slight  diminution  in  temperature 
was  observed  in  A's  calorimeter,  and  a  slight  rise  in  temperature 
in  B's  calorimeter.  With  the  usual  precautions  the  loss  of  heat 
in  A  and  the  gain  of  heat  in  B  were  calculated,  and  these 
quantities  were  found  to  be  equal. 

The  particles,  in  rushing  from  A  to  B,  needed  energy  to 
give  them  the  necessary  motion.  This  energy  was  taken  from 
the  heat  of  A,  with  a  consequent  fall  of  temperature.  The 
velocity  of  the  particles  was  retarded  again  in  B.  A  certain 
amount  of  momentum  was  destroyed,  the  result  being  the 
generation  of  heat,  and  the  temperature  of  B  rose. 

Joule  now  came  to  the  general  conclusion  that,  if  air  ex- 
pands without  doing  external  work,  there  is,  on  the  whole, 
neither  rise  nor  fall  in  temperature,  and  that  there  is  neither 
gain  nor  loss  of  energy  in  internal  work. 

Certain  theoretical  considerations  showed,  however,  that, 
save  in  the  case  of  a  perfect  gas  rigorously  obeying  Boyle's 
law,  the  loss  of  heat  in  one  case  was  not  exactly  equal  to 


272  Heat 

the  gain  in  the  other,  and  that  the  internal  work  could  be 
measured,  if  more  delicate  experiments  were  devised.  This 
led  to  the  following  arrangement. 

178.  Joule's  and  Thomson's  Experiments. — The 
object  of  the  research  is  stated  by  the  experimenters  in  the 
following  terms  * : — 

"  Let  air  be  forced  continuously,  and  as  uniformly  as 
possible,  by  means  of  a  forcing-pump,  through  a  long  tube, 
open  to  the  atmosphere  at  the  far  end,  and  nearly  stopped  in 
one  place  so  as  to  leave,  for  a  short  space,  only  an  extremely 
narrow  passage,  on  each  side  of  which,  and  in  every  other  part 
of  the  tube,  the  passage  is  comparatively  very  wide ;  and  let 
us  suppose,  first,  that  the  air,  in  rushing  through  the  narrow 
passage,  is  not  allowed  to  gain  any  heat  from,  nor  (if  it  had  any 
tendency  to  do  so)  to  part  with  any  to,  the  surrounding  matter. 
Then,  if  Mayer's  hypothesis  were  true,  the  air,  after  leaving  the 
narrow  passage,  would  have  exactly  the  same  temperature  as 
it  had  before  reaching  it.  If,  on  the  contrary,  the  air  ex- 
periences either  a  cooling  or  a  heating  effect  in  the  circum- 
stances, we  may  infer  that  the  heat  produced  by  the  fluid 
friction  in  the  rapids,  or,  which  is  the  same,  the  thermal 
equivalent  of  the  work  done  by  the  air  in  expanding  from  its 
state  of  high  pressure  on  one  side  of  the  narrow  passage  to  the 
state  of  atmospheric  pressure  which  it  has  after  passing  the 
rapids,  is  in  one  case  less,  and  the  other  more,  than  sufficient 
to  compensate  the  cold  due  to  the  expansion  ;  and  the  hypo- 
thesis in  question  would  be  disproved." 

Experiments  were  first  conducted  in  which  the  air  passed 
through  a  single  orifice ;  but  it  was  found  difficult  to  determine 
the  exact  temperature  of  the  issuing  air — its  temperature  varied 
with  the  distance  from  the  orifice,  part  of  the  energy  was  also 
used  in  imparting  kinetic  energy  to  the  particles  of  air,  and  in 
starting  sound-waves  due  to  the  air  rushing  through  the  orifice. 

"  The  porous  plug  was  adopted  instead  of  a  single  orifice, 
in  order  that  the  work  done  by  the  expanding  fluid  might  be 
immediately  spent  in  friction,  without  any  appreciable  portion 

1  Phil  Mag.,  1852.  Reprinted  in  Thomson's  and  in  Joule's  scientific 
papers. 


Thermo- Dynamics 


273 


of  it  being  even  temporarily  employed  to  generate  ordinary 
vis-viva,  or  being  devoted  to  produce  sound.  The  unconduct- 
ing  material  was  chosen  to  diminish  as  much  as  possible  all 
loss  of  thermal  effect  by  conduction,  either  from  the  air  on  one 
side  to  the  air  on  the  other  side  of  the  plug,  or  between  the 
plug  and  the  surrounding  matter." 

Along  tube  (Fig.  122)  has  a  diaphragm,  B  C  L  M,  of  cotton 
wool,  the  tube  is  surrounded  by  another,  R  S  T  U,  containing 
cotton  wool  or  some  other  bad  conductor,  to 
prevent  any  heat  entering  or  leaving  the  internal 
tube.  The  air  before  entering  this  protected 
part  of  the  tube  passes  through  a  copper  spiral 
76  feet  long.  The  spiral  is  surrounded  by 
water  kept  well  stirred,  whose  temperature  S 
is  read  carefully  from  thermometers  openly 
divided.  Thus  the  temperature  of  the  air 
when  it  meets  the  diaphragm  is  known.  A 
very  delicate  thermometer  is  placed  beyond 
B  C  to  determine  the  temperature  of  the  air 
after  it  passes  through  the  diaphragm. 

The  air  is  forced  through  the  tube  by  a 
piston  moving  uniformly  with  a  constant  pres- 
sure, P  (for  convenience  the  piston  P  is  shown 
in  the  figure ;  it  would  be  placed,  of  course, 
in  a  part  of  the  tube  beyond  the  spiral),  care 
being  taken  that  the  piston  moves  uniformly — 
there  is  therefore  no  increase  in  kinetic  energy 
— and  that  there  is  therefore  no  loss  of  energy 
due  to  sound-waves  caused  by  the  air  whirling 
through  the  diaphragm.  A  steady  flow  of  air 
passes  through  the  tube,  so  that  equal  masses 
will  cross  any  section  of  the  tube  in  equal  FIG.  122. 
times.  The  pressure  beyond  the  diaphragm  is  that  of  the 
atmosphere,  P'. 

We  are  noting,  not  the  effect  of  a  sudden  push  of  the  piston, 
but  of  a  continuous  uniform  movement  of  the  piston,  that  may 
continue  as  long  as  we  please.  This  gives  the  arrangement  one 
marked  advantage  over  that  in  §  177. 


274 

In  the  case  of  air  it  was  found  that  the  temperature  was 
lower  after  flowing  through  the  diaphragm  ;  the  lowering  varied 
with  the  differences  in  pressure.  Thus  when  P  and  P'  differed 
by  i  atmosphere,  the  temperature  fell  0*26°  C.  If  the 
difference  was  2  atmospheres,  the  fall  in  temperature  was  0-52°. 

Or  if  d  =  the  fall  in  temperature  in  C.°  j 

P  =  the  pressure  of  the  air  before  entering  the  plug  ; 
P'  =  atmospheric  pressure  on  the  other  side  of  the  plug, 
both  in  atmospheres; 

d 


,  =  o-26°C.  for  air 


P  -  P 

The  results  of  the  experiments  show,  for  air  and  similarly 
for  many  other  gases  (hydrogen  is  an  exception),  that  the  heat 
of  friction  in  the  porous  plug,  which  is  the  thermal  equivalent 
of  the  work  done  by  the  air  as  it  expands,  is  insufficient  to 
compensate  for  the  heat  required  to  produce  expansion,  and 
there  is  a  consequent  fall  in  temperature. 

If  the  pressure  on  the  piston  be  but  slightly  above  i 
atmosphere,  the  internal  work  in  the  case  of  air  is  only  about 

—  of  the  work  done  in  forcing  the  air  through  the  plug. 

In  the  case  of  carbonic  acid  gas  the  cooling  is  i'i5i°  for 
a  difference  of  i  atmosphere  at  20°  C. ;  at  91*5°  it  is  only  0703° 
per  atmosphere.  That  is,  the  internal  work  becomes  less  as 
the  gas  is  raised  above  its  critical  point. 

Hydrogen  is  an  exception.  There  is  a  slight  rise  in 
temperature  after  it  expands.  The  internal  work  is  negative. 

With  a  small  difference  of  pressure  it  is of  the  external 

work. 

Thus  we  see  that  air,  hydrogen,  and  similar  gases  may,  at 
ordinary  temperatures,  and  with  small  variations  in  pressure, 
be  treated  as  if  they  were  perfect  gases,  that  is,  that  the 
internal  work  done  by  the  gas  in  expanding  freely  is  practi- 
cally nil.  The  same  can  be  said  of  gases  like  carbonic  acid 
gas  and  sulphur  dioxide,  provided  their  temperature  is  suffici- 
ently high. 


T/ter  mo- Dynamics  275 

The  series  of  experiments  proved  conclusively  that  if  a  gas 
be  compressed,  the  work  done  in  compression  is  not  exactly 
equivalent  to  the  heat  produced  by  the  compression,  but  that 
the  internal  work  may  be  a  positive  or  negative  quantity,  or 
may  be  «//,  according  to  the  gas  experimented  upon,  and  that 
therefore  no  assumptions  must  be  made  as  was  done  by  Mayer. 

With  definite  knowledge  derived  from  experiment  of  the 
internal  work  of  air  and  other  gases  in  expansion,  and 
recognizing  that  it  is  so  small  in  certain  gases  as  to  be 
negligible  in  first  approximations,  we  can  safely  apply  Mayer's 
method  in  determining  the  mechanical  equivalent  of  heat 

179.  Determination  of  the  Internal  Work  of  Ex- 
pansion.—Collecting  data  from  experimental  results,  we 
obtain  the  following  : — 

Let  V  (AD EH)  and  V  (BCFG)  represent  the  volume  of 
i  Ib.  of  air  before  and  after  expansion  (Fig.  122); 
use  the  other  letters  P,  P'  as  explained  above. 

PV  =  the  work  done  by  the  piston  in  forcing  the  air  up  to 

the  plug  (see  §  186). 
P'V'  =  work  done  by  the  air  against  the  atmosphere  (see  p.  60). 

A  pressure  of  i  atmosphere  =147  Ibs.  per  square  inch 
=  2116-8  Ibs.  per  square  foot. 
If  W  =  total  work  done  in  expansion  in  foot-pounds— 

W  -  2 1 1 6-8(P'V  -  P  V)  =  work  spent  in  friction  in  the  plug  (F) 

W-2ii6-8(P'V'-PV).    ,  .       /T 

=  -  — y— in  heat-units    (J  =  1390) 

because  the  total  work  has  its  thermal  equivalent  (a)  in  the  heat 
of  friction  in  the  plug  (call  this  F),  together  with  (b)  the  excess 
of  the  work  done  by  the  air  against  the  atmosphere,  over  the 
work  done  in  forcing  the  air  up  to  the  plug. 

Let  H  =  the  actual  heat  necessary  to  compensate  for  the  cold 
of  expansion. 

Heat  of  friction  in  the  plug  (F)  =  H,  minus  the  heat  that  must 
be  supplied  to  keep  the  gas,  after  expansion,  at  the  same 
temperature  as  the  gas  before  expansion  (//) 


276  Heat 

For  i  Ib.  of  air,  h  =  i  x  specific  heat  of  air  at  constant  pres- 

sure X  d    \d=  o-26(P  -  P')] 
=  0-2375  X  o-26(P  -  P') 
=  o-o62(P  -  P') 

=      =       __  Q         p  _ 


1390 

.'.  H  =  JL  +  o-o62(P  -  P')  -  2Il6>8(P'v  _ 
1390  1390 

Neglecting  the  small  difference  P'V'  -  PV, 


1390 

The  total  work  done  when  a  gas  represented  by  (PV)  ex- 
pands to  (P'V) 

=  PV  logep-(§  1  86)  =  PV?-^?'  nearly 

if  the  difference  between  P'  and  P  be  small. 
This  work  in  foot-pounds  — 

w=  f 


Therefore  the  ratio  of  the  excess  of  the  actual  heat  needed 
in  expansion  over  the  thermal  equivalent  of  the  work  done  in 
expansion  equals  — 

H        W 


1390 

(P'=>  ^sphere) 


i  Ib.  of  air  at  o°  C.  and  i  atmosphere  pressure  occupies 
1  2  -39  cubic  feet.  Therefore,  if  the  experiment  be  performed 
when  the  temperature  is  15°  C.  — 

PV  at  15°  =  12-39  x  ~  =  13-1  nearly       * 

0-062  x  1390  =  0. 

2116-8  X  13-1 


Thermo-  Dynamics  277 

This  is  slightly  above  the  real  value  ;  we  have  neglected— 

(P'V  -  PV)2ii6-8 

1390 
With  this  correction— 

r  =  0-0024  for  air  =  — 
4i7 

Therefore  if  we  subject  atmospheric  air  to  a  pressure 
slightly  above  i  atmosphere,  at  ordinary  temperatures,  and 
if-  W  be  the  work  done  in  compression,  the  heat  evolved  is 
mechanically  equal  to  — 


If  carbon  dioxide  be  experimented  upon   under   similar 
conditions  — 


The  heat  evolved  would  be  mechanically  equal  to  W  +  — 

In  the  case   of  hydrogen,  the   heat   evolved  would  be 

W 

mechanically  equal  to  W  — 


1250 

180.  Determination  of  the  Mechanical  Equivalent 
of  Heat  from  Hydrogen.— (i)  Assume,  what  is  practically 
true,  that  hydrogen  in  expanding  behaves  like  a  perfect  gas, 
that  there  is  no  internal  work  done  by  it  or  upon  it  when  it 
expands  freely.  (Mayer's  hypothesis.) 

(2)  Facts  collected  from  experiments. 

(a)  i  gram  of  hydrogen   at  o°  C.  and  under  a  pressure  of 
760  mm.  of  mercury,  measures  n, 164-45  cc- 

(b)  The  pressure  due  to  a  height  of  760  mm.  of  mercury 
is    1033*3  grams   per   square    centimetre  in   the   latitude   of 
London. 

(c)  The  specific  heat  of  hydrogen  at  constant  pressure  (C) 
is  3*409 ;  the  specific  heat  at  constant  volume  (c),  as  deter- 
mined from  the  velocity  of  sound,  is  3 -409  4-1-41,  because 
C 


Heat 


Let  us  experiment  upon  i  kilogram  of  hydrogen  at  o°  C. 
and  760  mm.  pressure.  Its  volume  will  be  11*16445  cubic 
metres.  Take  as  the  unit  of  heat  the  heat  required  to  raise 
the  temperature  of  i  kilogram  of  water  i°  C. 

Enclose  the  hydrogen  in  a  cylinder,  as  in  §  176,  whose 
section  is  1-116445  square  metre,  then  the  gas  will  stand 
10  metres  high  in  the  cylinder.  The  pressure  on  the  piston 
will  be  10333  X  1*116445  kilogram  =  11536*2  kilograms. 

(1)  Allowing    the   gas   to   expand   at   constant  pressure, 
apply  heat  until  its  temperature  rises  i°.     This  will  require 
a  number  of  units  of  heat  numerically  equal  to  the  specific 

heat  at  constant  pressure  (C).     The  piston  will  rise  —  metre, 

and  do  —  X  11536*2  kilogrammetres  of  work  =  422*6  kilo- 
grammetres of  work. 

(2)  Beginning  with  the  gas  at  its  original  volume,  tempera- 
ture, and  pressure,  keep  the  piston  fixed,  and  apply  heat  to 
raise  the  temperature  i°.     This  will  require  c  units  of  heat. 
c  =  specific  heat  of  hydrogen  at  constant  volume. 


Heat- 
units. 

Dynamical  result. 

C 

c 

Temperature  of  I 
kilogram  raised 
i° 

Temperature  of  i 
kilogram  raised 
i° 

Internal  work. 

External  work. 

Practically       nil 
from  Joule's  and 
Thomson's   ex- 
periments 
Nil 

422*6      kilogram- 
metres 

Nil 

.'.  C  —  c  is  equivalent  to  422*6  kilogrammetres  of  work 

C 


also  c  = 


1*41 


1-41 


.*.  C  is  equivalent  to  422-6  x  — —  kilogrammetres 


Thermo-Dynamics  279 

.*.  if  J  be  the  mechanical  equivalent  of  i  unit  of  heat — 

_  422*6  x  1*41 
J  ~"  0-41  x  3*409 
=  426*2  kilogrammetres  of  work 

The  student  should  notice  that  the  specific  heat  at  constant 
volume  (c)  is  determined  from  the  specific  heat  at  constant 
pressure  (C)  by  the  relation — 

C 

7  =  i*4i 

This  ratio  is  frequently  calculated  from  relations  involving 
the  mechanical  equivalent  of  heat.  Evidently  c  calculated  in 
this  way  is  inadmissible  in  any  determination  of  the  mechani- 
cal equivalent  of  heat.  C  and  c  must  either  be  determined 

Q 

separately  or  —  be  determined  by  some  independent  method, 

such  as  the  method  based  on  the  velocity  of  sound  (§  183). 

181.  General  Case,  assuming  Mayer's  Hypothesis. 
— Let  V,  P,  T  represent  the  volume,  pressure,  and  absolute 
temperature  of  a  perfect  gas  whose  mass  is  i  gram. 

(i)  Let  the  mass  be  heated  to  a  temperature  T'°,  expanding 
at  constant  pressure  P. 

(a)  It  must  be  supplied  with  C(T'  — T)  thermal  units. 

T'        PV      PV' 

(b)  The  new  volume  V  becomes  V  •  —  (••  —  =  - 


/i   —  i  \ 

/.  the  increase  in  volume  =  V  •  ( — ^ — j 

And  the  external  work  done  =  PV 


r-rs 

~Y~) 

T'  -  T 


T 

(2)  Without  work  being  done  upon  it,  let  the  volume  V 
change  to  V,  the  temperature  remaining  at  T'°,  the  pressure  will 

V 
change  so  that  the  new  pressure  P'  =  P  •  y 

/PV      P'V\ 
VT7"  :  :  ~T7 

(3)  Let  heat  escape  until  the  pressure  becomes  P.     The 


280  Heat 

volume  being  V,  the  temperature  must  be  T ;  the  gas  is  losing 
heat  at  constant  volume. 

/.  it  loses  <r(T-T)  units 

The  gas  is  now  in  its  original  condition.     It  has  received 
C(Tf-T)   and  lost  r(T'-T)   units   of  heat,   and   has   done 

T'  —  T 
PV  •  — FF; —  units  of  external  work. 

T'  —  T 

/.  (C  -  c)  (T'  -  T)  units  of  heat  are  equivalent  to  PV jp— 

units  of  work 
If,  therefore,  J  is  the  mechanical  equivalent  of  heat — 

• i__    PV 

J  ~  C-^'  T 
PV      P0V0         -    C  i\         k-i 


PV      P0V0        .    C  i\ 

-TjT-  =  -7p-  and  if  -  =  /£,  C  -  c  =  C(i  -  ^  J  = 


C 


op 

•  •  J  -  k  -  r  c  '  T0 

k          £    P0V0 

"  k-  i*  C'  273 

£  for  gases  that  are  nearly  perfect  is  1*41  ;  P0  is  the  pressure 
due  to  760  mm.  of  mercury,  i.e:  a  pressure  of  1033-3  grams 
per  square  centimetre.  V0,  the  volume  of  i  gram  of  the 

gas  at  o°  C.  and  760  mm.  pressure,  equals  ^-,  where  D0  is  the 

*-*o 
density  under  the  same  conditions. 


T  =  . 

J  /-\  *   A    T  rt 


0-41      273      CI)0 


The  units  are  the  gram,  centimetre,  degree  Centigrade.  To 
change  into  the  usual  units— kilogram,  metre,  degree  Centi- 
grade— we  must  multiply  by 


1000 


1000  x  100 
,'.  J  =  °'i3°2 


Thermo-  Dynamics  2  8  1 

.*.  if  we  take  D  as  the  mass  of  i  cubic  metre,  then  — 

J  =  130200.^ 

For  air,  C  =  0*2375  ;  D0  =  1293. 

.'.  J  =  423*9  kilogrammetres  of  work 
For  oxygen,  C  =  0-2175  ;  D0  =  1429-8. 

/.  J  =  420 

The  determination  is  not  so  near  as  the  above,  but  for  oxygen 
k  -  1*403.     With  this  correction  J  =  425. 

For  nitrogen,  C  =  0*2438  ;  D  =  1256. 

•'•  J  =  425 

182.  The  Difference  in  Thermal  Units  of  the  Two 
Specific  Heats.  —  Returning  to  the  formula  where  C  —  c  is 

PV 

equivalent  to  TTT,  and  noting  that  — 

PV      P0V0 

-?JT  =  -Fp-  =  constant  R 

we  have  C  —  c  -  R. 

This  enables  us  to  determine  the  difference  between  the 
two  specific  heats. 

It  is  only  necessary  to  determine  R  under  the  most  con- 
venient conditions. 

For  example,  for  air  at  o°  C.  and  760  mm.  pressure  as 
before  — 

P0V0 

-7T-    =    2927 


The  thermal  equivalent  will  be     p  =  r~     =  0-0691 

.*.  C  —  c  =  0*0691 
Regnault  found  C  to  be  0-2375. 

.'.  c  -  0-2375  -  °'o69i  =  0-2684  (see  §  66) 

183.  Ratio  of  the  Specific  Heats  deduced  from  the 
Velocity  of  Sound.—  Newton  concluded  from  mathematical 


282  Heat 

investigations  that  the  velocity  of  sound  could  be  calculated 
from  the  formula  — 


where  v  =  velocity,  E  =  elasticity,  and  D  =  density.  In  air 
or  other  gas  at  ordinary  pressure,  the  elasticity  is  equal  to  the 
pressure  P. 


At  o°  C.  and  760  mm.  pressure,  the  pressure  is  equal  to 
I033'3  grams  per  square  centimetre.  The  accelerative  force  of 
gravity  =981  centimetres  per  second  per  second. 

/.  E  =  P  =  1033-3  X  981 
D  =  density  =  0-001293  gram  per  cubic  centimetre. 


.%  v  =  \J    ^^ — -  =  28,358  centimetres  per  second 
v    0-001293 

.Actual  experiment  makes  the  speed  about  33,000  centi- 
metres per  second.  Such  a  difference  between  calculation  and 
fact  showed  that  some  error  had  been  committed  in  the 
calculation. 

Newton,  in  his  calculation,  allowed  for  the  fact  that,  when  a 
gas  is  compressed,  its  elasticity  is  increased  because  its  density 
is  increased.  If  we  compress  a  gas  very  slowly,  so  that  the 
heat  is  allowed  to  escape,  the  temperature  does  not  rise,  and 
this  increase  of  elasticity,  due  to  increase  of  density,  is  the  only 
increase  to  allow  for.  But  by  the  fire-syringe  experiment  it 
is  seen  that,  if  the  gas  be  compressed  suddenly,  heat  is 
evolved,  and  this  heat  increases  the  elasticity — that  is,  an 
increased  pressure  must  be  applied  to  keep  its  volume  con- 
stant. Similarly,  if  a  gas  be  suddenly  rarefied,  the  gas  is  cooled, 
and  this  lowering  of  temperature  lessens  the  elasticity. 

What  takes  place  when  sound  travels  in  air  ?  The  particles 
are  suddenly  compressed  at  the  condensed  part  of  the  wave, 
and  are  therefore  heated.  The  particles  are  suddenly  separated 
at  the  rarefied  part  of  the  wave,  and  are  cooled  as  much  as  they 
were  heated  at  the  condensed  part;  therefore  the  average  tern- 


Thermo- Dynamics  283 

perature  remains  the  same.  Has  this  any  effect  on  the  speed 
of  the  wave  ?  The  condensed  particles  are  heated,  their  elas- 
ticity is  increased ;  they  therefore  tend  the  more  to  separate, 
they  separate  the  more  rapidly,  and  thus  the  speed  of  the  wave 
is  increased.  Condensation  is  succeeded  by  rarefaction,  and 
the  rarefied  part  is  cooled.  Its  elasticity  is  reduced ;  therefore 
the  condensed  particles  in  front  are  the  more  easily  able  to 
rebound,  and  again  the  speed  of  the  wave  is  increased. 

The  condensation  and  rarefaction  take  place  so  rapidly  that 
heat  has  not  time  to  escape  by  conduction. 

Taking  these  facts  into  account,  Laplace  showed  that  the 
formula  for  the  velocity  must  express  the  elasticity  under  the 
conditions ;  that  is,  real  elasticity  =  E ;  and  that  the  corrected 
formula  was — 


/  E  /C     P 

"=  V   D=  V    7'D 


—  a  formula  that  includes  the  ratio  of  the  two  specific  heats. 
C  D 


z>,  D,  and  P  can  be  measured,  and  therefore  ~  the  ratio    - 

can  be  determined. 

The  ratio  is  found  to  be  equal  to  1-41  (§  66). 

184.  Adiabatic  Lines.  —  The  relation  between  the 
volume  (v)  and  pressure  (/)  of  a  perfect  gas  while  the  tem- 
perature remains  constant  is  — 

pv  =  a  constant 
or  pv  =  RT 

where  T  is  the  absolute  temperature.  This  relation  is  repre- 
sented graphically  by  an  isothermal  line,  the  curve  being  a 
rectangular  hyperbola  (§  42). 

The  formula  is  only  approximately  true  in  experiments 
if  the  variation  in  volume  takes  place  slowly,  so  that  the  heat, 
the  result  of  compression,  has  time  to  escape.  In  many 
operations  we  approximate  to  a  condition  in  which  the  heat- 


284 


Heat 


energy  cannot  enter  or  escape  from  the  apparatus,  as,  for 
example,  when  we  compress  or  allow  a  gas  to  expand  suddenly, 
or  when  we  compress  or  allow  a  gas  to  expand  in  a  vessel 
whose  sides  are  bad  conductors.  There  is  in  such  experiments 
a  rise  or  fall  in  temperature  (§  183). 

Let  A  (Fig.  123)  represent  the  relation  between  volume, 
pressure,  and  temperature  of  a  given  mass  of  a  gas  at  any 


D' 


FIG.  123. 

given  time,  the  volume  being  O  vy  the  pressure  v  A  or  O/, 
and  the  temperature  T.  T  A  C  D  is  the  isothermal  for  the 
given  mass  of  the  gas  for  T° ;  T'  B  D'  is  the  isothermal  for 
the  same  gas  at  some  higher  temperature,  T'°. 

If  the  pressure  be  increased,  so  that  heat  cannot  escape,  the 
temperature  generally  will  rise,  and  there  will  be  increased  resist- 
ance to  compiession,  so  that  a  greater  increase  in  pressure  will 
be  required  to  effect  a  given  decrease  in  volume  than  would  be 
the  case  if  the  heat  generated  were  allowed  to  escape.  Thus 
if  we  take  the  gas  when  the  absolute  temperature  is  T'°  and 
the  volume  O  z/,  the  pressure  O/'  or  v'  B  will  be  greater  than 


Thermo- Dynamics 


285 


\ 


O/r  or  z/  C,  the  pressure  corresponding  with  the  volume  O  z/ 
when  the  temperature  remains  constant 

This  reasoning  will  apply,  however  small  the  difference 
between  T°  and  T'°.     The  new  curve,  AB,  called  an  adia- 
batic  curve,  expresses  the  relation  between  the  volume  and 
pressure  of  a  gas  when  no   heat 
is  allowed  to  enter  or  escape;  when 
it  crosses  an  isothermal  curve,  it 
makes  a  greater  angle  with  the  line 
of  volumes  than  does  the  isothermal 
curve  (Fig.  124). 

In  the  isothermal  lines  the  tem- 
perature is  constant;  in  the  adia- 
batic  lines  there  is  also  a  quantity 
that  is  constant ;  this  will  be  dealt  with  in  §  196. 

In  the  above  we  have  discussed  the  adiabatic  curve  with 
respect  to  a  substance  that  remains  in  the  gaseous  condition. 
The  line  can  be  drawn  equally  well  for  liquids  and  solids,  save 
that  as  in  the  case  of  the  isothermals  the  diminution  of 
volume  is  so  small  compared  with  the  increase  in  pressure,  that 
it  is  not  easy  to  represent  the  variation  on  a  diagram. 

The  adiabatic  for  a  mixture  of  vapour  and  liquid  is  im- 


ISOTHERMAL 


•-AD/ASAT/C 


FIG.  124. 


FIG.  125. 

portant.     Let  us  take  the  best  known  case  of  steam  and  water. 

Let  the  position  A  (Fig.  125)  represent  the  volume  and 

pressure  of  i  Ib.  of  steam  at  212°  F.     From  Regnault's  tables 


286  Heat 

we  know  that  the  pressure  will  be  14*7  Ibs.  per  square  inch, 
and  the  volume  will  be  26*36  cubic  feet.  -  So  that  on  the 
diagram  a  A  =  26*36,  and  Oa  -  147. 

(1)  If  we  keep  the  temperature  constant  and  attempt  to 
decrease  the  volume,  we  find  tha.  the  pressure  remains  constant, 
part  of  the  steam  condenses ;  in  fact,  we  describe  the  isother- 
mal A  a  for  steam  in  the  presence  of  its  liquid,  and  during 
the  process  part  of  the  heat  must  escape.      If  from  A  we  keep 
the  temperature  constant  and  increase  the  volume,  then  the 
steam  remains  a  vapour,  acts  nearly  like  a  gas,  and  the  part 
of  the  isothermal,  A  b,  is  described. 

(2)  If  the  i  Ib.  of  steam  be  in  a  cylinder  closed  with  a 
piston,  all  impervious  to  the  flow  of  heat,  then,  if  the  pressure 
be  increased,  the  temperature  of  the  steam  will  rise,  and  allow 
it  still  to  exist  as  steam  at  the  higher  pressure.     For  example, 
if  the  temperature  rises  to  213°  R,  the  steam  will  remain  as  a 
vapour  provided  that  the  pressure  is  14*99  Ibs.   Per  square 
inch;  with  a  temperature  of  215°  the  steam  will  only  begin  to 
condense  if  the  pressure  be  more  than  15*6  Ibs.  on  the  square 
inch.     The  actual  pressure  in  this  experiment  is  not  sufficient 
to  produce  condensation,  and  thus  the  adiabatic  line  is  a  curve, 
A£     It  can  also  be  shown  that,  if  the  steam  be  in  contact 
with  water  at  the  same  temperature,  increase  in  the  pressure 
produces  sufficient  heat  not  only  to  allow  the  steam  to  exist 
as  steam,  but  to  evaporate  part  of  the  water. 

Returning  to  the  point  A,  let  the  volume  increase,  heat 
being  prevented  from  entering  or  leaving  the  substance.  The 
steam,  expanding,  moves  the  piston  and  does  work.  There  is 
a  consequent  fall  in  the  temperature ;  the  fall  is  so  great  that 
when  the  pressure  is  say  14  Ibs.  on  the  square  inch,  the 
temperature  to  keep  all  as  steam  should  be  209°  F.  ;  it  is, 
however,  lower  than  this,  so  that  part  of  the  steam  condenses, 
the  liberation  of  heat  keeping  the  remainder  dry  and  saturated. 
For  any  given  pressure  the  volume  will  be  less  than  it  would 
be  for  an  isothermal ;  the  adiabatic  curve,  A  d,  will  fall  below 
the  isothermal  A  b. 

b  A  a  is  the  isothermal  for  steam  at  212°  F. 

is  the  adiabatic  curve  for  steam  passing  through  A; 


Thermo- Dynamics  287 

the  point  A  represents  the  condition  that  the  temperature  is 
212°  F.,  the  pressure  147  Ibs.  per  square  inch,  and  the 
volume  26-36  cubic  feet. 

The  isothermal  curve  for  a  perfect  gas  is  a  rectangular 
hyperbola,  whose  equation  is — 

pv  -  a  constant 
or   pv  =  RT 

where  T  is  the  absolute  temperature. 

The  value  of  R  is  determined  from  one  observation.  For 
example,  in  the  case  of  i  Ib.  of  air  at  o°  C.  and  at  normal 
pressure  (2116*8  Ibs.  per  square  foot),  the  volume  is  12 "39 
cubic  feet,  and  the  constant  is — 

R  =  pv  =  2116-8  x  12-39  =    6 
T  273 

In  an  adiabatic  curve  increase  of  volume  is  accompanied 
by  greater  decrease  of  pressure  than  would  be  the  case  along 
any  isothermal  passing  through  the  same  point. 

It  can  be  shown  that  generally  the  relation  between  pressure 
and  volume  along  an  adiabatic  curve  can  be  expressed  by — 

pvn  -  constant 

Q 

For  a  perfect  gas  n  =  —  =  1-41  (§  183) ;  that  is — 
pv  r41  =  a  constant 

17 

For  dry  steam  an  approximate  formula  is  pv&  -  a  constant 
185.  Work  done  by  a  Gas  in  Expanding. — If  a  gas  be 
subjected  to  a  given  uniform  pressure,  as,  for  example,  that  of 
the  atmosphere,  then,  if  the  gas  expands,  it  does  work  against 
this  pressure.  The  measure  of  the  work  done  will  be  the 
pressure  into  the  distance  moved.  In  the  case  of  a  cubic  foot 
of  gas  at  atmospheric  pressure,  let  it  expand  to  i^  cubic  foot ; 
then  the  increase  in  volume  is  ^  cubic  foot.  If  the  movable 
piston  be  \  square  foot  in  area,  it  will  move  i  foot. 

The  pressure  on  i  square  foot  =  14-7  x  144  =  2116 '8  pounds 
i         „          =  2116-8  x^ 
Work  done  =  (2116-8  x  i)  X  i  foot-pounds 
=  axi6'8x(iX  i) 


288 


Heat 


^  X  i  is  the  increase  in  volume,  so  that  work  done  = 
pressure  X  increase  in  volume. 

Generally,  if  the  pressure  be  P  per  unit  area,  and  the  area 
of  the  piston  be  A,  the  total  pressure  is  PA,  if  the  distance 
moved  be  x. 

Work  done  is  PA*  =  P  X  Ax  =  Pv 
where  v  is  the  increase  in  volume. 

186.  Indicator  Diagram.— Graphic  Representation 
of  Work  done. — Let  Oa  (Fig.  127)  represent  a  volume,  #  A 
a  pressure,  according  to  a  definite  scale. 


FIG.  126. 


a        b        c  v 

FIG.  127. 

Let  the  gas  expand  from  O  a  .to  O  £,  the  pressure  remain- 
ing uniform  ;  that  is,  the  line  A  B  will  be  horizontal. 

The  work  done  =  pressure  X  increase  in  volume 
=  0A  X  (Ob  -  Oa) 
=  #  A  X  ab 
=  area 


If  the  pressure  increases  or  decreases  uniformly,  it  is  still 
easy  to  estimate  the  work  done.  Let  the  initial  pressure  and 
volume  be  B  b,  O  b.  Let  the  volume  increase  to  O  c  while  the 
pressure  decreases  uniformly  to  C  c9  so  that  B  C  is  a  straight 
line.  As  the  pressure  has  changed  uniformly,  we  can  take 
the  average  pressure 

=  KB*  +  O) 
Work  done  =  P  x  increase  in  volume 


area 


Thermo-  Dynamics 


289 


Generally  the  line  is  a  curve,  A  B  (Fig.  128).  We  then 
estimate  the  work  done  in  expanding  from  O  a  to  O  b  by 
dividing  a  b  into  a  very 

\ 


large  number  of  equal 
parts,  and  drawing  the 
verticals  cm,  dn,  etc. 

If  a  c  —  c d  .  .  .  be 
very  small,  no  appreci- 
able error  will  be  made 
by  calling  A  ;//,  m  ;/, 
.  .  .  straight  lines.  The 
work  done  will  then  be 
the  areas  a  A  me,  cm  n  d, 
...  or  the  total  work 
the  area  of  the  figure 
a  A  B  b. 


V 


a  fid 


The    estimation    of  FIG.  128. 

such  areas  belongs  to  pure  mathematics,  and  it  can  generally 
be  done  if  the  form  of  the  curve  A  B  is  known. 

The  two  curves  of  most  frequent  use  in  heat  are  pv  =  con- 
stant, and/?'"  =  constant. 

( i )  Area  of  a  A  B  b  when  pv  =  constant. 

If  the  point  A  represents  pj\,  while  B  represents/-/'.. — 


The  area 


or  = 


Ob 
X  Oa  x  loge  Q^ 

Ob 
xObx  log£  rr- 


That  is— 


7'o  7'.> 

Work  done  =  p$\  loge  --  =  p.rc»  log€  _," 


(T  is  absolute  temperature) 

?o  7'o 

.'.  work  done  =  /V'o  loge  ~"  —  R*  ^°oe ,.," 


<  i 


(2)  Area  ofa\Rb  when pi>*  =  constant. 

X  Oa  -  Kb  X  Ob 

n  —  I 


Area  = 


290 


Heat 


;.  work  done  = 


11  —~ 


A  frequent  value  of  n  is  —  =  1*41. 


work  done  = 


0*41 


By  the  aid  of  an  indicator  diagram  we  can  estimate  the 
work  done  by  or  on  a  working  substance  (a)  by  calculating, 
if  the  curve  be  a  known  curve ;  or  (f)  by  measuring  the  area. 

Attached  to  steam-engines  is  an  apparatus  that  registers 
automatically  the  variations  in  volume  and  pressure. 

Let  A  be  the  starting-point,  and  let  the  curve  traced  be 
A  B  C  D  E.  From  A  to  C  the  working  substance  does  work, 
whose  measure  is  the  area  a  A  B  C  c.  From  C  to  E  work  is 


FIG.  129. 

done  on  the  substance;  its  measure  is  cCEe;  from  E  to  A 
the  substance  does  work,  whose  measure  is  e  E  A  a. 

Work  done  by  =  eE\a  +  aABCc 
„  upon  =  cCEs 

:.  nett  work  done  =  ABCEA 


TJiermo-Dynamics  291 

Returning  to  the  point  A  means  that  the  working  sub- 
stance is  again  in  exactly  the  same  physical  condition  as  at 
starting. 

In  many  operations  this  condition,  that  the  working  sub- 
stance is  brought  into  its  original  condition  as  regards  volume, 
temperature,  and  pressure  (only  two  need  be  denned),  is  not 
fulfilled,  but  in  the  discussion  that  follows  this  is  always  the 
case,  so  that  the  curve  is  closed. 

187.  A  Cycle  of  Operations. — If  a  series  of  operations 
be  performed  upon  a  working  substance,  so  that  its  final  physical 
state  is  in  all  respects  similar  to  its  original  state,  the  series  of 
operations  is  called  a  cycle  of  operations. 

For  example,  we  may  begin  with  a  pound  of  steam  at 
212°  F.  Its  pressure  will  be  147  Ibs.  on  the  square  inch,  while 
its  volume  will  be  26-36  cubic  feet.  We  may  increase  the 
pressure,  preventing  the  escape  of  heat,  when  we  shall  form 
the  adiabatic  curve  for  steam,  and  the  temperature  will  rise. 
Let  the  new  pressure  be  20  Ibs.  on  the  square  inch.  Then  we 
may  allow  heat  to  escape  until  the  temperature  falls  to  228°  F. 
If  a  further  reduction  of  temperature  takes  place  with  this 
pressure,  some  of  the  steam  will  condense.  Imagine  that  the 
temperature  sinks  to  200°.  All  the  steam  will  have  condensed, 
and  if  the  pressure  remains  at  20  Ibs.  on  the  square  inch,  the 
water  will  be  compressed. 

Now  let  heat  be  applied  until  the  temperature  is  2i2°F., 
the  pressure  remaining  at  20  Ibs.  per  square  inch.  Reduce 
the  pressure  to  147  Ibs.  on  the  square  inch  ;  the  whole  will 
evaporate,  and  its  volume  will  be  26*36  cubic  feet  as  before. 

The  final  condition  is  exactly  the  same  as  the  original 
condition,  both  as  regards  volume,  pressure,  and  temperature. 
A  cycle  of  operations  has  been  performed. 

Another  example  may  be  cited.  Unit  mass  of  air  is  taken 
at  certain  volume  (z>),  temperature  (/),  and  pressure  (/>). 

(1)  Heat  is  supplied,  the  volume  remaining  constant  until 
the  temperature  rises  to  t°  and  the  pressure  becomes  tf  (i.e. 
c(f  —  /)  units  of  heat  supplied). 

(2)  The  mass  is  allowed  to  expand,  doing  work,  until  the 
pressure  is  again  /,  the  temperature  remaining  t10. 


292 


Heat 


(3)  Heat  is  taken  away  until  the  temperature  falls  to  /°, 
the  volume  becoming  again  v  (C(/  —  t)  units  taken  away). 

(4)  The  mass  is  now  in  the  original  condition  (v,  /,/). 
A  cycle  of  operations  has  been  performed. 

188.  Garnet's  Cycle. — Carnot,  in  1824,  described  a 
theoretic  heat-engine  that  has  been  of  great  service  in  the 
study  of  heat.  Carnot  believed  in  the  then  accepted  idea  that 
heat  was  a  material  substance,  and  in  the  modern  statements 
his  original  method  is  altered  to  suit  the  dynamical  theory. 
Carnot's  memoir  was  neglected  until  its  value  was  pointed 
out  by  Sir  W.  Thomson. 

The  engine. — An  essential  part  of  the  engine  is  that  the 
working  substance  may  be  any  substance  whatever.  The 
results  are  independent  of  the  substance  ;  it  may,  for  example, 
be  air,  steam,  hydrogen,  a  liquid,  a  solid,  or  partly  liquid  and 
vapour,  etc.  It  is  convenient  to  take  a  gas,  as  the  variations 
in  volume  are  more  marked ;  but  this  selection  is  a  matter 
purely  of  convenience. 

The  substance  D  (Fig.  130)  is  contained  in  a  cylinder  so 


s 

A 
HOT 


FIG. 


constructed  that  the  sides  and  the  piston  are  perfect  non- 
conductors, so  that  no  heat  can  enter  or  leave  by  them.  The 
capacity  for  heat  of  the  piston  and  sides  is  zero.  The  base  of 
the  cylinder  is  a  perfect  conductor. 

The  other  parts  are  (i)  a  hot  body,  A,  called  the  source, 
which  is  kept  at  a  constant  temperature,  S ;  (2)  a  body,   B, 


Thermo- Dynamics 


293 


colder  than  A,  which  is  kept  at  a  constant  temperature,  T,  and 
which  is  called  the  refrigerator ;  and  (3)  a  stand,  C,  the  upper 
surface  of  which  is  a  perfect  non-conductor,  like  the  sides  of 
the  cylinder  and  the  piston. 

Let  the  cylinder  be  placed  on  A  when  the  temperature  of 
the  working  substance  is  S;  then,  whatever  motion  of  the 
piston  takes  place,  since  heat  passes  freely  through  the 
base,  and  the  supply  from  S  is  inexhaustible,  the  relation 
between  volume  and  pressure  will  always  be  expressed  by 


FIG. 


the  isothermal  of  the  substance  for  S  degrees  (curve  AS, 
Fig.  131).  The  point  A  indicates  the  condition  that  the 
pressure  (/\)  is  A  a,  the  volume  (z\)  O#,  and  the  tempera- 
ture S. 

If  the  cylinder  be  placed  on  B  when  the  temperature  of 
the  working  substance  is  T,  then  the  relation  between  volume 
and  pressure  will  always  be  expressed  by  the  isothermal  for  T°. 

The  point  C  gives  for  the  same  mass,  a  volume  (v^  =  Qc, 


294 


Heat 


pressure  C  c  (/3),  and  temperature  T  (the  student  will  note  that 
no  particular  scale  of  temperature  is  defined). 

If  the  cylinder  be  placed  on  the  stand,  then,  as  no  heat  can 
enter  or  escape,  the  relation  between  volume  and  pressure  will 
be  expressed  by  an  adiabatic  line ;  the  particular  line  depend- 
ing upon  the  original  conditions  of  volume,  temperature,  and 
pressure  (Fig.  132). 

For  the  purpose  we  have  in  view  the  adiabatic  curves  pass 


FIG. 132. 

through  the  points  A  and  C  of  Fig.  131,  which,  we  have  seen, 
represent  the  relation  of  pressure,  volume,  and  temperature 
as  described  in  the  figure. 

Let  Fig.  131  be  placed  on  Fig.  132.  Then  the  points  A  and 
C  coincide,  also  the  adiabatics  and  isothermals  intersect  at  B 
and  D,  and  the  value  of  volume  and  pressure  can  be  calculated 
for  these  two  positions  (Fig.  133). 

Suppose  that  for  the  given  working  substance  the  isothermals 
S  and  T  have  been  carefully  determined,  also  that  the  adia- 
batics passing  through  A  and  C  have  been  determined ;  that 


TJienno-Dyna  in  ics 


295 


is,  that  by  preliminary  experiments,  Fig.  133  has  been  mapped 
out  Furthermore,  we  may  imagine  that  indicators  register 
on  this  perfectly  theoretical  machine  the  pressure  and  volume, 
and  also  the  quantities  of  heat  taken  in  or  given  out  at  any 
period.  Carnot's  four  operations  can  now  be  followed. 

First  operation. — Place  the  cylinder  on  the  hot  body  A,  and 
begin  the  operation  when  the  pressure  is/!,  the  volume  z>lf  and 
the  temperature  S.  This  condition  is  indicated  on  the  diagram 
by  the  point  A  (Fig.  133). 

Now  let  the  substance  expand.    Heat  flows  from  S,  keeping 


a  ft        6  c 

FIG.  133. 

the  temperature  constant  at  S,  therefore  the  isothermal  curve 
A  B  is  described.  When  the  volume  is  O  b  (z-2)  and  the  pressure 
b  B  (/2),  stop  the  operation.  A  quantity  of  heat,  which  we  may 
call  H,  has  entered  the  working  substance. 

Second  operation. — Place  the  cylinder  on  the  stand.  Let  the 
body  continue  to  expand.  It  will  do  so  adiabatically,  and  the 
relation  between  the  pressure  and  volume  will  be  represented 


296  Heat 

by  the  curve  B  C.  The  temperature  gradually  falls.  When 
it  reaches  the  temperature  T,  stop  the  operation,  and  place  the 
cylinder  on  the  cold  body  B.  So  far  the  working  substance  has 
been  doing  work  by  moving  the  piston  against  external  pressure. 

Third  operation. — Begin  to  push  back  the  cylinder,  that  is, 
do  work  on  the  working  substance.  Variation  in  temperature 
is  impossible,  seeing  that  heat  readily  escapes  to  B.  The 
isothermal  C  D  is  described.  When  the  position  is  reached 
expressed  by  D  (volume  ?'4,  pressure  /4),  stop  the  operation ; 
that  is,  stop  at  a  point  selected  so  that  the  compression 
in  the  fourth  operation  will  bring  the  substance  back  to  its 
original  conditions.  A  quantity  of  heat,  //,  has  escaped  from 
the  working  substance  to  B.  Place  the  cylinder  on  the  stand. 

Fourth  operation. — Continue  to  press  back  the  cylinder. 
Heat  is  generated,  the  temperature  rises,  and  the  adiabatic 
curve  D  A  is  described,  When  the  temperature  T  is  reached, 
the  position  will  be  A ;  that  is,  the  pressure  is  /lf  the  volume 
is  zfj,  and  the  temperature  is  S. 

The  working  substance  is  now  exactly  in  the  original  con- 
dition. A  cycle  of  operations  has  been  performed. 

The  determination  of  the  points  A,  B,  C,  D,  is  evidently 
possible. 

Let  us  sum  up  what  has  been  done. 

(1)  In  the  first  and  second  operations  a  quantity  of  heat, 
H,  has  entered  the  working  substance,  and  the  amount  of  work 
done  by  the  substance  is  measured  by  the  areas  A  B  b  a,  B  C  c b. 

(2)  In  the  third  and  fourth  operations  a  quantity  of  heat,  //, 
has  been  taken  from   the  working  substance,  and  the  work 
done  on  the  substance  is  measured  by  c  C  D  d  and  d  D  A.  a. 

A  quantity  of  heat,  H,  at  temperature  S,  has  been  taken  in, 
and  a  quantity,  //,  at  temperature  T,  has  been  given  out. 

The  excess  of  work  done  by  the  substance  over  the  work 
done  in  it  is  — 

ABC^-C^AD 

that  is,  the  work  measured  by  A  B  C  D. 

That  is,  a  quantity  of  heat,  H  — //,  is  equivalent  to  a  quantity 
of  work  that  is  measured  by  A  B  C  D. 


Thermo-  Dynamics  297 

If  H  and  h  are  measured  in  heat-units,  and  J  is  the 
mechanical  equivalent  of  heat  — 

J(H-//)  =  area  ABCD  -  W  the  useful  work 

If,  as  is  generally  done,  we  express  H  or  h  in  dynamical 
units  (foot-pounds  or  foot-poundals)— 

H-/I  =  W 

189.  Illustration  from  a  Perfect  Gas.  —  If  the  working 
substance  be  a  gas  obeying  Boyle's  law,  and  we  measure 
temperature  from  absolute  zero  on  the  Centigrade  scale,  we 
obtain  the  following  :  — 

If  A  represent  /^S;  B,/^S;  C,/,raT;  D,/4r4T— 

Work  done  =  ABCD  =  AB&*  +  BGtf  -  cCVd  - 
(See  §  186.)      =  p^  loge     +^_^  _  /<r>  loge 


—    I 


/^  =  RS,  etc.     Loge  ^  -  loge  |r 

/.  ABCD  -  RS  loge  ?f-  RT  loge  %  +  -     -  (S  -  T  -  S  +  T) 

t'  t.       i»  •  •  i 


The  efficiency  of  a  machine  is  the  ratio  of  the  work  done 
to  the  heat  supplied. 

w      Rtog.(S-T)      S_T 


H  =  R  loge  —  '  S  ;  because  it  equals  external  woik  done, 

f'i 

since  the  internal  work  is  zero  (§§  178,  179). 
For  example,  if  S  be  100°  C.  and  T  o°  C.— 

The  efficiency  = 


374  o74 


298  Heat 

With  a  given  temperature  of  supply,  S,  we  can  only  increase 
the  efficiency  by  obtaining  a  lower  temperature  for  the 
condenser. 

The   solution    of    the    problem    of    obtaining    increased 

S  —  T 
efficiency  lies  in  making  — ^ —  as  nearly  as  possible  equal  to 

unity ;  that  is,  in  making  T  as  nearly  as  possible  equal  to  o° 
on  the  absolute  scale,  or  in  making  S  as  large  as  possible. 

The  student  should  carefully  note  that  in  this  illustration 
S  and  T  are  temperatures  on  the  Centigrade  scale  measured 
from  absolute  zero  ;  in  §  1 88  we  avoided  defining  the  scale  of 
temperature. 

190.  The  Reversible  Cycle. — The  important  feature  of 
Carnot's  cycle  is  that  it  is  reversible.  Instead  of  taking  the 
order  A  B  C  D,  it  can  be  worked  in  the  order  A  D  C  B. 

First  operation. — Let  the  condition  of  the  working  substance 
be  represented  by  the  point  A.  Pressure  =  /a,  temperature  = 
S,  volume  =  v^. 

Place  the  cylinder  on  the  non-conducting  stand,  and  let 
the  substance  expand;  it  will  do  so  adiabatically,  and  the 
temperature  will  fall.  When  the  temperature  falls  to  T,  the 
volume  will  be  v&  and  the  pressure  p4.  These  conditions  will 
be  represented  on  the  diagram  by  D.  Remove  the  cylinder 
to  the  cold  body,  whose  temperature  is  T. 

Second  operation. — Let  the  substance  still  expand;  it  will 
do  so  at  constant  temperature,  and  the  variations  of  tempera- 
ture and  pressure  will  be  expressed  by  the  isothermal  D  C. 
When  the  volume  is  z>3  and  the  pressure  p.A,  stop  the  operation, 
and  remove  the  cylinder  to  the  non-conducting  stand. 

During  this  operation  a  quantity  of  heat  has  entered  the 
working  substance.  It  is  equal  to  the  quantity  that  flowed 
from  it  in  the  cycle  (§  188),  when  the  volume  was  changed 
from  v.2  to  z>3,  that  is,  the  quantity  is  h. 

Third  operation. — Press  back  the  piston  when  the  cylinder 
is  on  the  stand.  Ths  variations  of  temperature  and  pressure 
will  be  expressed  by  the  adiabatic  curve  C  B.  When  the 
temperature  is  S  and  the  volume  v±  and  pressure  /o,  remove 
the  cylinder  to  the  hot  bod)-. 


Thermo- Dynamics  299 

Fourth  operation. — Continue  to  press  back  the  cylinder. 
The  temperature  remains  constant  at  S,  and  a  quantity  of 
heat,  H,  enters  the  hot  body  from  the  working  substance. 
When  the  pressure  is  pl  and  the  volume  vlt  stop  the  operation. 
The  cycle  is  now  complete,  as  the  working  substance  is  in 
exactly  the  same  condition  as  at  starting. 

Work  done  by  Work  done  on 

the  substance  the  substance 

measured  by  measured  by 

First  operation  ADda 

Second    „  

Third       „  ... 

Fourth     „  

The  nett  work  done  on  the  substance  is  measured  by  the 
area  A  B  C  D. 

To  effect  this  work  h  units  of  heat  have  been  taken  from 
the  colder  body,  and  H  units  have  been  given  to  the  hot 
body. 

h  -  H  =  -  ABCD  =  -  W 
or       H  -  h  =  W 

That  is  the  same  relation  as  in  the  direct  action.  We  see 
that  in  this  reversible  engine  a  quantity  of  heat,  H,  at  a  given 
temperature,  S,  can  enter  the  engine,  and  a  quantity,  //,  can  be 
given  out  at  a  lower  temperature,  T,  with  the  result  that  useful 
work,  W,  is  done ;  or  a  given  quantity  of  heat,  //,  at  a  tempera- 
ture, T,  can  enter  the  engine,  and  a  quantity,  H,  can  be  given 
out  at  higher  temperature,  S,  if  work  measured  by  W  be  done 
upon  the  engine. 

191.  A  Reversible  Engine  the  most  Efficient.— Of 
all  engines  working  between  a  temperature,  S,  and  a  lower 
temperature,  T,  and  receiving  a  quantity  of  heat,  H,  at  the 
higher  temperature,  the  reversiole  engine  is  the  most  efficient. 

W 

The  efficiency,  we  have  seen,  is  — : 

H 

This  proposition  is  proved  by  joining  two  engines.  A  is 
a  reversible  engine  working  between  S  and  T,  taking  in  heat  H 
at  S,  giving  out  heat  h  at  T,  and  doing  useful  work  W.  B  is 
another  engine  working  between  the  two  temperatures  S  and 


300  Heat 

T,  taking  in  heat  H  at  S,  and  doing,  if  possible,  more  work, 

W  +  Z£/. 

Let  a  quantity  of  heat,  H,  from  the  hot  body  enter  the 
engine  B.  As  it  is  more  efficient  than  the  engine  A,  the  work 
done  is  greater  than  W ;  let  us  call  the  work  done  W  -j-  w. 
As  it  has  transferred  a  greater  quantity  of  heat  into  work,  it 
will  give  up  a  less  quantity  to  the  cold  body ;  call  this  amount 
h  —  //'.  Let  the  work  W  be  used  to  work  the  engine  A  in 
the  reverse  order.  This  work  will  take  a  quantity  of  heat,  //, 
from  the  cold  body,  and  give  H  units  to  the  hot  body. 

The  total  result  is  that  the  hot  body  gives  up  and  receives 
H  units ;  that  is,  it  does  not  lose  heat. 

The  cold  body  receives  h  —  H  and  gives  up  h  units ;  that 
is,  it  loses  h1  units. 

The  useful  work  is  (W  +  w)  -  W  =  w. 

This  can  be  repeated  as  often  as  we  please,  and  thus  the 
whole  of  the  heat  might  be  taken  from  a  body  by  this  com- 
bined engine  without  any  further  assistance,  and  changed  into 
work. 

This  work  is  now  available,  and  might  be  used  in  the  re- 
versible engine  working  in  the  reverse  order,  to  raise  the  heat  of 
a  body  at  any  temperature  to  a  higher  temperature,  and  thus 
ultimately  all  the  heat  of  the  universe  might  be  changed  to 
heat  at  a  higher  temperature. 

Suppose  a  refrigerator  were  at  o°  C,  and  all  surrounding 
objects  at  100°  C.  If  a  reversible  engine  were  not  perfect,  then 
it  would  be  possible,  by  simply  cooling  the  refrigerator  below 
o°  C.,  to  do  mechanical  work  ;  or  it  would  be  possible  to  raise  a 
portion  of  the  heat  from  o°  to  heat  at  100°  C,  and  allow  this  to 
flow  back  to  heat  at  o°,  and  in  so  doing  produce  mechanical 
work. 

This  is  contrary  to  experience,  and  therefore  we  conclude 
that  our  hypothesis,  namely,  that  any  engine  can  have  a 
greater  efficiency  between  two  given  temperatures  than  a 
reversible  engine,  is  false. 

A  reversible  engine  shows  a  maximum  efficiency  ;  that  is, 
out  ot  no  other  engine  working  between  the  same  temperatures 
can  more  work  be  obtained  for  the  same  supply  of  heat,  and 


Thermo-Dy  'nam  ics  30 1 

it  follows  that  all  reversible  engines  are  equally  efficient ;  that 
is,  the  efficiency  is  independent  of  the  working  substance. 

192.  The  Second  Law  of  Thermo-dynamics.— From 
such  reasoning  the  Second  Law  of  Thermo-dynamics  is  deduced. 
The  law  is  expressed  in  various  forms — 

"  It  is  impossible  for  a  self-acting  machine,  unaided  by  any 
external  agency,  to  convey  heat  from  one  body  to  another  at  a 
higher  temperature." 

"  It  is  impossible  by  means  of  inanimate  material  agency  to 
derive  mechanical  effects  from  any  portion  of  matter  by  cooling 
it  below  the  temperature  of  the  coldest  of  the  surrounding 
objects." 

193.  Thomson's  Scale  of  Temperatures.— Thomson 
has  shown  that  Carnot's  cycle  leads  to  a  method  of  defining 
temperature.     In  ordinary  measurements  we  determine  tem- 
perature from  the  expansion  of  mercury  or  other  liquid  inside 
glass,  and,  as  has  been  seen,  while  two  such  thermometers  agree 
at  certain  points,  they  do  not  agree  at  intermediate  points, 
unless  the  same  kind  of  glass  and  the  same  liquid  be  used. 

The  regularity  in  the  expansion  of  air  at  constant  pressure 
led  to  the  selection  of  air  or  other  gas  as  a  thermometric  sub- 
stance. It  is  seen  that  for  such  gas — 

r,  -  rt,(i  +  at) 

If  we  wish  to  determine  by  calculation  at  what  tempera- 
ture the  volume  =  o  (we  have  no  knowledge  of  such  a  tem- 
perature, and  are  not  sure  what  changes  would  take  place  in  a 
gas  for  some  degrees  above-  such  a  temperature, — certainly 
most  of  them  would  be  liquefied  (§  123),  perhaps  solidified), 
we  have — 

?'t  =  o  =  v»(i  +  at) 
:.    i  4-  at  =  o 

i 

~  a 

a  is  found  by  experiment  to  be  0-003665°  on  the  Centigrade 
scale. 

•*'  /=  -  0-003665  =  -272 '85°  C;  approximately,  -273°  C. 


3O2  Heat 

This  temperature  is  called  the  absolute  zero  of  the  Centi- 
grade scale.  The  determination  of  the  zero  is  dependent 
upon  a  particular  substance — air,  hydrogen,  etc. 

Any  temperature  on  this  scale  T  =  /  +  - 

The  scale  suggested  by  Thomson  is  independent  of  the 
working  substance.  In  §  189  it  is  shown  that  if  the  working 
substance  be  a  perfect  gas,  and  temperature  be  measured  from 
the  absolute  zero  as  deduced  from  the  expansion  of  air, 

W      S  —  T 

then  the  efficiency  =  g  =  — ^ — ;    that  is,   the  efficiency  is 

S  —  T 
proportional  to  — g — ,  an  expression  in  which  temperature 

only  is  involved. 

Carnot  showed  that  this  is  true  for  any  reversible  engine  if 
the  difference  of  temperature  be  very  small,  so  that  the  differ- 
ence is  a  very  small  fraction  of  one  degree.  This  can  be 
extended  to  prove  that  generally  the  efficiency  of  a  reversible 
engine  depends  on  the  temperature  of  the  source  and  refrigera- 
tor alone,  and  is  independent  of  the  working  substance. 

Carnot's  proof  can  be  applied  to  ordinary  Centigrade 
degrees  as  measured  on  an  ordinary  thermometer. 

Thomson,  assuming  that — 

W      S  -  T 
H  "       S 
H- h      S-T 
°r    -H-  =  -S~ 
H      S 

or  J  =  T 

is  rigorously  exact,  and  is  from  Carnot's  cycle  independent  of 
the  working  substance,  denned  temperature  thus — 

The  temperatures  of  two  bodies  are  proportional  to  the 
quantities  of  heat  respectively  taken  in  and  given  out  in  loca- 
lities at  one  temperature  and  at  the  other  respectively  by  a 
material  system  subjected  to  a  complete  cycle  of  perfectly 
reversible  thermo-dynamic  operations,  and  not  allowed  to  part 
with  or  take  in  heat  at  any  other  temperature.  Or — 


Thermo-  Dynamics  303 

The  absolute  values  of  two  temperatures  are  to  one  another 
in  the  proportion  of  the  heat  taken  in  and  the  heat  rejected  in 
a  perfect  thermo-dynamic  engine,  working  with  a  source  and 
refrigerator  at  the  higher  and  lower  temperatures  respectively. 

S      H 
T=l 

S  and  T  are  temperatures  as  defined  on  Thomson's  thermo- 
dynamic  scale.  H  and  h  are  measurable  quantities. 

The  size  of  the  degrees  is  a  matter  of  convenience,  as 
is  also  the  zero  of  the  scale.  It  is  therefore  admissible  to 
attempt  to  adapt  such  a  theoretic  scale  to  our  ordinary  scales  ; 
for  example,  between  melting  point  of  ice  and  boiling  point  of 
water  in  the  Centigrade  scale  there  are  100  divisions  ;  in  the 
thermo-dynamic  scale  we  can,  similarly,  make  100  divisions. 

It  is  beyond  the  scope  of  this  work  to  give  all  the  steps  in 
the  calculation  of  the  relation  between  the  thermo-dynamic 
scale  and  the  Centigrade  scale. 

The  experimental  work  is  that  detailed  in  §  178.  It  was 
concluded  by  Thomson  that  — 


0  =  temperature  on  thermo-dynamic  scale. 

/  =         ,,  „      Centigrade  scale. 

a  =  coefficient  of  expansion  of  the  gas. 

K  =  specific  heat  of  the  gas   at   constant  pressure  in 

dynamical  units. 
T  =  increase  in  temperature  after  the  gas  passes  through 

the  plug  (§  178). 
S  =  constant  for  gas  =  o/Vv 
A  /'  ~  pressures  on  each  side  of  the  plug. 

In  the  experiments  T  was  very  small  (see  also  §  178),  and, 
save  in  the  case   of  hydrogen,  was   negative.     In  all  cases 

£  •  -      -.  was  small  :  let  it  be  indicated  by  y,  a  small  variable 
*>     i       P 

los«/ 

quantity. 


304  Heat 

Now,  /  +  -  =  temperature  measured  from  absolute  zero  of 
the  Centigrade  scale  =  T. 

/.  6  =  T  ±  y  for  all  gases 

Thus  the  thermo-dynamic  scale  is  obtained  by  adding  or 
subtracting  a  small  variable  quantity  to  the  Centigrade  scale 
measured  from  absolute  zero. 

In  one  series  of  experiments   the  following  results  were 
obtained  :  — 
Pressure  of  air  forced  into  the  plug  (/)  =  20*943  Ibs.  per  sq.  inch 

„          atmosphere  (/')  =  I4'777          „          „ 

Temperature  of  air  before  entering  plug  =  16°  C. 
Cooling  effect  (T)  =  0-105°  C. 

K  =  0-2375  x  1390  =  330-125  foot-pounds 
a  =  0*003665 


^0  =  14-7  x  144  =  2116-8  Ibs.  per  square  foot 
?70  =  volume  of  i  Ib.  of  air  at  o°  C.  and  2116*8  Ibs.  per 
square  foot  pressure  =  12-39 

.*.  a/0z-0  =  0-003665  x  2116-8  x  12-39  =  96*122 
P  P 

- 


I  KT 

'+;  -T 


.r         .. 

96-12  Iog2o'943-log  14777 

,00  o   ,   34-663     0-4343 

=  2oo  o  H  --  s  ---  •  - 
9O-I2   0-15145 

=  288-8  +  1-03 
=  289-83 

The  difference  between  the  number  of  degrees  on  the 
Centigrade  and  thermo-dynamic  scales,  from  freezing  point 
to  1  6°  C.,  will  be  very  small.  Therefore  we  conclude  that 


Thermo-Dynainics  305 

freezing  point  on  the  thermo-dynamic  scale  is  approximately 

273'830. 

Careful  calculations  from  many  experiments  showed  that 
if  the  difference  in  temperature  between  melting  point  of  ice 
and  boiling  point  of  water  be  represented  by  100°,  then  the 
zero  of  the  thermo-dynamic  scale  is  —  273*7°;  if  the  difference 
in  temperature  be  represented  by  180°,  the  zero  is  —460-66°. 

It  is  assumed  that  /Or0  =pMv^\  this  is  not  exactly  true, 
and  recognizing  this  difference  would  introduce  a  correction 
into  the  formula. 

Centigrade  scale  on 

Thermo-dynamic  air-thermometer 

scale.  (constant  volume). 

2737+0°  o°  (freezing  point) 

+  20°  20  +  0-0298° 

+  40°        ...  40  +  0-0403° 

+  60°       60  +  0-0366° 

+  80° 80  +  0-0223° 

+  100°  100  +  0°  (boiling  point) 

+  120°  ...          ...  120  —  0*0284° 

+  200°  ...         ...  200  —  o  1798° 

194.  Freezing  Point  of  Water  lowered  by  Pres- 
sure.— This  was  first  deduced  by  Professor  James  Thomson, 
from  theoretical  considerations.1  The  following  are  the 
essential  steps  :  Let  a  mass  of  air  be  enclosed  in  a  cylinder 
(the  air-cylinder)  similar  to  that  described  in  §  188;  let  the 
air  be  compressed,  and  the  temperature  be  o°  C.  :  the  pressure 
on  the  piston  we  can  call  p.  Let  a  small  quantity  of  water 
be  enclosed  in  an  exactly  similar  cylinder  (the  water-cylinder), 
let  the  pressure  be  that  of  the  atmosphere,  and  the  tempera- 
ture also  o°  C.  Place  these  cylinders  together,  with  their 
bases  in  contact,  the  two  forming  an  engine ;  heat  will  readily 
pass  through  the  bases,  but  cannot  escape  from  the  engine 
as  a  whole,  since  the  sides  and  pistons  are  perfect  non- 
conductors. 

Let  the  piston  of  the  air-cylinder  move,  the  compressed 

1  Transactions  of  the  Royal  Society  of  Edinburgh,  1849.     Reprinted  in 
Sir  "William  Thomson's  papers. 


306  Heat 

air  doing  work,  until  the  pressure  falls  to  i  atmosphere. 
Heat  will  be  needed ;  this  will  flow  from  the  water  at  o°,  and 
therefore  part  of  it  will  freeze ;  in  freezing,  water  expands ; 
therefore,  if  the  water-piston  remain  fixed,  the  pressure  on 
the  piston  will  become  more  than  i  atmosphere.  The 
water  is  freezing  under  increased  pressure,  and  we  wish  to 
determine  its  temperature  while  freezing.  By  allowing  the 
water-piston  to  move,  the  water  does  work  ;  let  it  expand 
until  the  pressure  falls  to  i  atmosphere  ;  we  have  then  water 
and  ice  at  o°  C.  Whatever  the  temperature  may  have  been 
when  freezing  began,  this  will  also  be  the  final  temperature 
of  the  air-cylinder. 

Now  press  the  air-cylinder  back  at  o°  C.  until  the  air  attains 
its  original  volume,  and  consequently  its  original  pressure ; 
the  heat  due  to  compression  will  flow  through  the  bases,  melt 
the  ice,  and  the  water-piston  will  be  acted  upon  by  the 
pressure  of  the  atmosphere,  which  will  do  work  on  the  piston. 
The  heat  of  compression  of  the  air  will  be  that  absorbed  in 
expansion,  and  therefore  the  whole  of  the  ice  will  again  be 
melted;  the  water  will  be  at  o°  C. ;  the  pressure  is  i  atmo- 
sphere, and  it  will  therefore  occupy  its  original  volume.  Both 
air  and  water  are  in  their  original  condition,  and  therefore 
any  work  done  by  the  engine  cannot  come  from  the  working 
substances. 

The  work  done. — The  distance  traversed  by  each  piston  is 
the  same  in  each  operation.  The  water-piston,  in  doing  work, 
begins  at  a  pressure,  as  we  have  seen,  greater  than  i  atmo- 
sphere, and  ends  with  a  pressure  of  i  atmosphere ;  the 
mean  pressure  is  therefore  greater  than  i  atmosphere.  When 
work  is  done  on  it,  the  pressure  throughout  is  i  atmosphere ; 
therefore  the  water-piston,  on  the  whole,  does  a  certain 
positive  amount  of  work.  This  can  only  come  from  the  work 
done  on  the  air-piston ;  therefore  more  work  must  be  done  on 
the  air-piston  in  compression  than  it  does  during  expansion. 
This  is  only  possible  if  the  temperature  during  compression 
is  greater  than  during  expansion;  during  compression  it  is 
throughout  at  o°  C,  in  expansion  it  is  finally  at  oc  C.,  there- 
fore when  expansion  began  it  must  have  been  below  o°  C. ; 


TJienno-Dynam  ics 


307 


but  this  was  the  temperature  of  the  ice  freezing  under  pressure, 
and  hence  \ve  infer  that,  when  ice  freezes  under  pressure,  its 
freezing  point  is  lowered. 

195.  To  Calculate  the  Lowering  of  the  Freezing 
Point, — The  last  section  shows  that,  in  a  mixture  of  ice  and 
water,  the  pressure  will  remain  constant  as  long  as  the  tem- 
perature does  not  change — it  will  be  the  pressure  corresponding 
to  the  given  temperature ;  the  isothermal  for  any  degree  will 
therefore  be  parallel  to  the  line  of  volumes. 

In  a  Carnot's  cycle,  let  the  working  substance  be  a  mass 


A 


\ 


FIG.  134. 

of  ice  which,  if  melted,  will  form  rather  more  than  i  cubic 
foot  of  water,  and  let  it  be  originally  subjected  to  a  pressure 
of  i  atmosphere ;  its  temperature,  therefore,  will  be  o°  C. 
The  temperature  of  the  source  S  is  o°  C.,  that  of  the  refrige- 
rator T  is  slightly  below  o°  C. 

First  operation. — Place  the  cylinder  on  the  source,  and  let 
the  piston  do  work  on  the  ice,  with  the  result  that  part  of  it 
melts.  When  i  cubic  foot  of  water  is  formed,  stop  the  opera- 
tion. The  volume  has  decreased  by  0*087  cubic  foot,  because 


308  Heat 

i  cubic  foot  of  water  at  o°  C.  =  i  '087  cubic  foot  of  ice  at  o°. 
The  first  operation  will  be  represented  graphically  by  AB 
(Fig.  134).  A  B  is  the  isothermal  for  the  mixture  at  S° 
(i.e.  o°  C).  AB  =  0-087. 

Second  operation. — Place  the  cylinder  on  the  non-conduct- 
ing stand,  and  force  the  piston  down ;  the  temperature  will  fall 
and  the  pressure  will  increase ;  the  adiabatic  curve  B  C  will 
be  described.  When  the  temperature  is  —f  C.,  the  tem- 
perature of  the  refrigerator,  stop  the  operation. 

Third  operation. — Remove  the  cylinder  to  the  refrigerator 
T,  and  let  the  working  substance  do  work  on  the  piston  ;  the 
volume  will  increase,  but  the  pressure  and  temperature  will 
remain  constant,  and  the  isothermal  C  D  for  T  (i.e.  —  /°  C.) 
will  be  described.  Stop  the  operation  when  the  volume  is  such 
that  it  is  represented  by  D,  that  is,  when  the  isothermal  crosses 
the  adiabatic  curve  passing  through  A. 

Fourth  operation. — After  placing  the  cylinder  on  the  stand, 
let  it  further  expand ;  it  will  do  so  adiabatically.  When  the 
temperature  is  that  of  the  source  S,  the  pressure  will  be  i 
atmosphere,  and  the  temperature  will  be  o°  C.  The  working 
substance  is  now,  in  all  respects,  in  exactly  the  condition  in 
which  it  started,  and  the  whole  process  is  reversible. 

The  work  done  is  represented  by  the  area  A  B  C  D  ;  the 
isothermals  are  very  near  to  each  other ;  and  the  area,  neglect- 
ing very  small  differences,  will  be  the  rectangle  whose  base  is 
B  A,  and  vertical  height  P  Q. 

/.  work  done  =  BA  x  PQ 

BA  =  0*087  f°ot ;  PQ  =  P  =  difference  in  pressure  in  pounds 
per  square  foot. 

.*.  work  done  =  o'oS'jp  foot-pounds 
But  in  a  cycle — 

Work  done     _  difference  of  temperature  between  S  and  T 
Heat  absorbed  ~  temperature  of  source 

temperature  being  on  the  absolute  scale. 

To  melt  i  cubic  foot  of  ice,  80  X  62-5  =  5000  thermal 
units  are  needed. 


Thcrmo-Dynamics  309 

S-T  =  /"(:.;    S=  2737 
.*.  work  done  in  dynamical  units  =  5000  x  1390  x  —     -  ft.-lbs. 

»  =  0-00000343^ 


If  we  express/  in  atmospheres  (P),  we  have  the  pressure 
of  i  atmosphere  =  144  X  147  Ibs.  per  square  foot  =  2116-8. 

/.  t  =  0*00000343  x  2ii6'8P 
=  0-0073? 

If  the  pressure  be  increased  by  i  atmosphere,  the  tem- 
perature of  freezing  will  be  lowered  by  about  0*0073°  C.;  and 
if  the  pressure  be  increased  to  about  137  atmospheres,  the 
temperature  will  be  lowered  i°  C. 

By  similar  reasoning,  it  can  be  shown  that  the  freezing 
points  of  wax,  lead,  silver,  and  substances  generally  that 
expand  in  liquefaction,  are  raised  by  increased  pressure. 

196.  Adiabatic  Curves.  —  Entropy.  —  Reference  was 
made  in  §  184  to  the  fact  that,  as  in  isothermal  curves  the  tem- 
perature was  constant,  so  in  adiabatic  curves  there  was  a 
constant  quantity.  The  condition  of  adiabatic  expansion  or 
contraction  is  that  no  heat  shall  enter  or  leave  the  working 
substance;  of  course,  seeing  that  during  adiabatic  change, 
work  is  done  by  or  on  the  working  substance,  there  will  be 
a  loss  or  gain  of  heat-  energy  due  to  transformation  of  energy. 
We  have  seen  that,  in  Carnot's  engine,  working  between  two 
temperatures,  S°  and  T°  (measured  on  the  thermo-dynamic 
scale),  if  Hx  and  H2  be  the  heat  taken  in  and  given  out 
respectively  at  the  two  temperatures  — 


Hx  is  the  heat  taken  from  the  source  at  temperature  S° 
(the  isothermal  is  A  B,  Fig.  133);  H2  is  the  heat  given  to 
the  refrigerator  at  temperature  T°  (the  isothermal  is  C  D). 

Between  A  B,  C  D  we  may  have  any  number  of  isothermals 
crossing  B  C  in  points  E,  F,  G,  H,  etc.  (the  student  can  insert 


310  Heat 

these  lines).  In  passing  from  the  adiabatic  B  C  to  the 
adiabatic  A  D,  along  the  isothermal  through  E,  F,  G,  .  .  . 
there  will  be  an  amount  of  heat,  H2',  given  out  at  tempera- 
ture T2°. 

By  the  above  equation  — 

Hj        Ho        H2 
"S"  =  T~  :=  17 

Therefore,  in  passing  from  any  adiabatic  curve,  A  D,  along 
an  isothermal,  to  another  adiabatic  curve,  B  C,  the  ratio  of 
heat  taken  in  or  given  out,  to  the  temperature  at  which  it  is 
taken  in  or  given  out,  is  constant. 

TT 

For  this  reason  a  quantity  of  the  form  ~r  is  regarded  as 

TT 

a  characteristic  quantity  of  a  body.     If  >p  remains  constant, 

the  body  is  changing  adiabatically,  or  along  an  adiabatic  curve 

H  . 
<jf>  =  ™  is  a  constant. 

This  quantity  $  is  called  the  entropy. 

H 


We  are  generally  dealing  with  differences  of  entropy,  and 
therefore  the  absolute  zero  of  entropy,  which  would  be  its 
condition  when  deprived  of  all  heat,  need  not  be  known  ; 
it  is  sufficient  to  select  any  arbitrary  zero. 

Clerk  Maxwell  reckons  the  entropy  from  a  standard  state 
defined  by  a  standard  temperature  and  pressure. 

"  The  entropy  of  a  body  in  any  other  condition  is  then 
measured  thus  :  Let  the  body  expand  (or  contract)  without 
communication  of  heat  till  it  reaches  the  standard  temperature, 
the  value  of  which,  on  the  thermo-dynamic  scale,  is  T.  Then 
let  the  body  be  kept  at  the  standard  temperature,  and  brought 
to  the  standard  pressure,  and  let  H  be  the  number  of  units 
of  heat  given  out  during  this  process.  Then  the  entropy 

TT 

of  the  body  in  its  original  state  is  —  ." 


Thermo-Dynamics  3 1 1 

Let  <^  =  — J  =  entropy  of  a  body  A  at  T^0 
i    -Ho B       T  ° 


Let  heat  pass  from  the  body  at  the  higher  temperature,  Tj0, 
to  the  body  at  the  lower  temperature,  T2°  ;  let  h  be  the  quantity 
of  heat  that  passes. 

The  entropy  of  A  is  now  <£/  =  —  ~  — 


»          » 


M 


/.  the  gain  of  entropy  =  (<£/  +  <£/)  -  (^  +  <£,)  =  A  -  A 


This  is  necessarily  positive,  as  T!  is  greater  than  To. 

Considering  a  series  of  such  changes  among  a  system  of 
bodies  A.  B,  C,  D,  .  .  .  we  see  that  there  will  always  be  a  gam 
in  entropy  ;  that  is,  the  entropy  of  a  system  always  tends 
to  a  maximum  ;  and,  generally,  the  entropy  of  the  universe 
tends  to  a  maximum. 

Referring  to  Carnot's  cycle,  the  isothermals  can  be 
represented  by  ^  and  4  and  the  adiabatics  by  <h  and  <£..,. 
Passing  from  ^  to  </>o  along  tlt  the  heat  taken  in  is  t^fa  —  <&). 
Passing  from  £,  to  </>a  along  /2,  the  heat  given  out  is  4(<£2  -  &). 

.*.  the  work  done  equals  the  difference  =  (/x  —  /2)  (<^3  —  </>j)  = 
area  ABC  D 


EXAMPLES.    XII. 

1.  Write  out  the  first  law  of  thermo-dynamic?,  and  state  how  it  was 
deduced. 

2.  Point  out  the  defect  in  Mayer's  method  of  determining  the  mechanical 
equivalent  of  heat. 

3.  Compare  the  methods  of  Joule,  and  Joule  and  Thomson,  in  testing 
Mayer's  hypothesis.     What  are  the  advantages  of  the  latter  method? 

4.  How  can  the  ratio  of  the  two  specific  heats  be  determined  from 


3 1 2  Heat 

the  velocity  of  sound  in  air  ?     Explain  Laplace's  correction  of  Newton's 
formulae, 

5.  What  are  the  essential  points  about  isothermal  and  adiabatic  lines  ? 

6.  Explain  and  illustrate  "a  cycle  of  operations." 

7.  Why  is  there  the  need  for  the  limitations  in  the  third  of  Carnot's 
operations  ? 

8.  Prove  that  a  reversible  engine  is,  of  all  engines,  the  most  efficient. 

9.  How   did    Sir  William   Thomson   determine   his   thermo-dynamic 
scale?     In  what  respects  does  it  differ  from  the  Centigrade  scale  (a)  of 
an  air-thermometer,  (b]  of  a  mercurial  thermometer  ? 

10.  What  do  you  mean  by  the  efficiency  of  a  steam-engine?     What  is 
the  theoretical  efficiency  of  a  steam-engine  \vhose  boiler  is  at  150°  C., 
and  whose  condenser  is  at  40°  C.  ? 

11.  A  definite  volume  of  steam,  on  entering  the  cylinder  of  a  steam- 
engine,  possesses  a  definite  amount  of  heat,  which  is  in  part  communicated 
to  the  cylinder,  and  in  part  carried  away  by  the  steam  after  it  has  done 
its  work  in  the  cylinder.     Supposing  all  the  heat  thus  communicated  and 
carried  away  to  be  collected,  would  it  or   would  it  not  be  equal  to  the 
heat  possessed  by  the  steam  immediately  before  entering  the  cylinder? 
If  there  be  a  difference,  what  is  the  cause  ? 

12.  Show  from  Carnot's  cycle  that  the  temperature  of  fusion  of  wax 
will  be  raised  by  pressure. 

13.  A  certain  heat-engine  is  supposed  to  be  perfect :  if  the  temperature 
of  the  source  is  280°  C.,  and  that  of  the  refrigerator  is  120°  C.,  find  the 
efficiency.     If  250  heat  units  are  supplied  by  the  boiler,  find  the  work 
done  in  foot-pounds. 

14.  A  i5OO-lb.  shot  strikes  a  target  with  a  velocity  ot  looo  feet  per 
second  :  if  all  the  kinetic  energy  be  transformed  into  heat,  and  all  the  heat 
generated  be  utilized  in  heating  1000  Ibs.  of  water  at  o°  C. ,  find  the  final 
temperature. 


3'3 


CHAPTER   XIII. 
A  PPL  fCA  T10NS—CLIMA  TE. 

197.  General. — In  the  preceding  chapters  reference  has 
been   made  to  many  applications   of  the   theory  of  heat  to 
climate,  ocean  and  air  circulation,  heating  by  hot  water,  etc. ; 
a  complete   explanation,  however,  was   impossible  until   the 
general  theory  of  heat  was  understood. 

198.  Heating   of  Buildings.— Open   Fireplaces. — 
Heat,  the  result  of  the  combustion  of  coal,  is  radiated,  and 
warms  the  room.     Combustion  is  made  possible  by  the  fresh 
air  passing  over  the  burning  coals  and  bringing  the  necessary 
supply  of  oxygen.      The  air   in    the  chimney  is  heated;   it 
becomes  less  dense  than  a  similar  column  of  colder  air  at  the 
temperature  of  the  room,  and  circulation  ensues. 

The  velocity  of  the  draught  depends  upon  the  height,  and 
therefore  the  chimney  should  be  high  enough  to  produce 
sufficient  draught  for  any  required  purpose.  The  chimney 
should  not  be  wider  than  necessary,  otherwise  there  is  a 
tendency  for  descending  currents  to  form.  The  air  entering 
the  chimney  should,  for  the  best  combustion,  pass  over  the  fire ; 
this  is  attained  by  making  the  entrance  to  the  chimney  small, 
and  just  above  the  fire.  The  effect  of  the  sheet  of  iron  that  is 
sometimes  placed  in  front  of  the  fire  is  to  make  all  the  air  pass 
through  the  fire ;  the  combustion  is  thereby  improved.  Bright 
metal  or  tiled  plates  serve  to  reflect  the  heat,  while  the  fire-brick 
back  becomes  hot  and  acts  as  a  radiator.  The  pleasant  feelings 
associated  with  a  bright  fire,  and  the  effective  aid  such  a  fire 
is  to  ventilation,  makes  the  open  fire  popular ;  it  is,  however, 


314  Heat 

far  from  economical,  and  probably  not  more  than  10  per  cent, 
of  the  heat  of  combustion  is  utilized. 

Gas  fires  in  some  cases  take  the  place  of  open  fires.  The 
heat  is  more  under  control,  and,  as  the  stove  generally  projects 
into  the  room,  the  metal  body  and  the  pipe  leading  to  the 
chimney  become  hot,  and  serve  as  radiators.  The  heated 
gases  become  well  cooled  before  entering  the  chimney,  and  the 
economy  is  therefore  greater.  Gas  fires  frequently  produce 
an  unpleasant,  irritating  feeling,  on  account  of  the  tempera- 
ture of  the  air  of  the  room  becoming  so  high  that  it  becomes 
very  "  dry."  This  can  be  remedied  by  keeping  vessels  filled 
with  water  in  the  room,  so  as  to  keep  the  air  nearer  its  point 
of  saturation.  In  comparing  the  relative  efficiency  of  gas  and 
coal,  it  must  be  remembered  that  gas  is  a  more  expensive 
form  of  obtaining  heat. 

199.  Hot-Water  Apparatus. — The  boiler  is  placed  as 
low  in  the  building  as  possible,  and  should  be  lower  than  any 
of  the  pipes.  The  feed-pipe  should  enter  the  bottom  of  the 
boiler,  and  the  pipe  that  conducts  the  hot  water  should  leave 
at  the  highest  point ;  this  outflow  pipe  should  be,  as  far  as  is 
practicable,  vertical.  The  heated  water  ascends  (§  138)  the 
vertical  pipe,  the  column  being  less  dense  than  the  column  in 
the  return-pipe.  During  its  return  the  water  passes  through 
suitable  iron  coils,  heats  these  coils,  and,  by  radiation,  heat  is 
communicated  to  the  various  apartments.  A  cubic  foot  of 
water  weighs  about  6  2^-  Ibs. ;  therefore,  in  cooling  one  degree, 
62 \  thermal  units  will  be  liberated.  One  cubic  foot  of  air 
weighs  0*0807  lb.,  and  its  specific  heat  is  0-2375,  therefore 
0*0807  x  0*2375  =  0*0192  thermal  unit  is  needed  to  heat 
i  cubic  foot  of  air  through  one  degree,  and  therefore  i 
cubic  foot  of  water  cooling  one  degree  will,  theoretically,  heat 
62'5 -{r  0*0192  =  3250  cubic  feet  of  air  through  one  degree. 
Knowing,  then,  the  supply  of  air  that  must  be  heated,  we  can 
determine  the  size  of  pipes  required,  and  the  amount  of  water 
that  must  pass  through  per  minute  at  a  given  temperature, 
to  properly  heat  any  room. .  Rooms  should  be  kept  at  from 
54*  to  68°  F.  during  the  day.  The  useful  heat  obtained 
will  depend  upon  the  temperature  of  the  fire,  the  surface  of 


Applications —  Climate  3 1 5 

the  pipe,  and  also,  as  questions  of  conductivity  and  radiation 
will  be  involved,  upon  the  nature  of  the  pipe.  As  the  radiator, 
iron  (or,  better,  copper)  will  be  a  suitable  material ;  the  pipes 
joining  the  radiators  should  be  covered  by  some  non-conducting 
substance,  such  as  felt,  if  such  pipes  are  not  required  for  heating 
purposes. 

200.  Heating  by  Steam. — In  heating  with  hot  water, 
the  high  specific  heat  of  water  is  a  great  advantage,  seeing 
that  the  water  cools  slowly.     The  system  is  of  great  advantage 
for  a  steady  supply  of  heat  at  a  temperature  below  100°  C. 
The  efficiency  of  a  radiator  is  greatly  increased  by  raising  the 
temperature.    This  can  be  attained  by  using  steam ;  by  heating 
the  steam  under  pressure  (§  117)  the  temperature  can  be  raised 
above  100°  C.,  and  the  radiators  can  be  kept  at  any  desired 
temperature  by  increasing  the  pressure.     Smaller  pipes  can  be 
used  for  connecting  the  radiators,  and  thus  less  heat  will  be 
lost  by  radiation  from  the  connecting  pipes.     If  the  pressure 
of  steam  ceases,  then  the  amount  of  water  condensed  is,  com- 
pared with   the  amount  that  would  be  present  in  hot-water 
pipes,  small,  and  therefore  the  temperature  will  rapidly  fall, 
despite  the  heat  liberated  in  condensation,  unless  the  fires  be 
attended  to.     This  is  in  many  cases  an  advantage,  because, 
as  soon  as  no  further  heat  is  required,  there  ceases  to  be  a 
waste  of  energy. 

201.  The  Heat  of  the  Sun. — The  principal  source  of 
heat,  as  we  have  seen  in  chapter  xi.,  is  the  sun.     The  radiant 
energy  passes  through  dry  air  almost  undiminished  in  quantity. 
Water  vapour  and  the  dust-particles  absorb  certain  light-rays, 
and  give  rise  to  rain-bands  in  the  visible  spectrum  ;  they  also 
have  an  absorptive  effect  upon  the   heat-rays.     The  radiant 
energy  at  the  surface  of  the  earth  is  changed  into  sensible  heat, 
and  warms  the  surface,  and  the  heat   is  again  radiated  into 
space.    It  is  now,  however,  proceeding  from  a  source  at  a  com- 
paratively low  temperature,  and  the  water  vapour  has  a  greater 
absorptive  effect  upon  such  radiation ;  therefore  the  earth  will 
lose  less  heat  when  surrounded  by  air  containing  vapour  than 
when  surrounded  by  comparatively  dry  air.     Probably  a  greater 
effect  is  due  to  the  condensed  water-particles  and  the  dust- 


3i6  Heat 

particles  that  have  been  shown  to  be  always  present  in  the  air. 
Clouds  serve  also  as  reflectors,  and  again  prevent  loss  of  heat 
into  stellar  space. 

202.  Dust. — The  action  of  dust  in  climatology  has  been 
investigated  by  Mr.  John  Aitken,  who  has  shown  that  dust- 
particles  can  always  be  detected  in  the  atmosphere.     They  are 
due  to  the  meteorites  that  reach  the  atmosphere — cosmic  dust 
— and  to  the  dust  carried  into  the  atmosphere  by  winds,  or 
projected  into  it  by  volcanoes.     Dust-particles  are  most  fre- 
quent over  towns,  and  decrease  as  we  reach  pure  mountain 
air.     The  result  of  Mr.  Aitken's  experiments  is  to  show  that 
water  vapour  cannot  condense  unless  dust-particles  be  present, 
and  that   therefore   condensation  is    impossible  in  an  atmo- 
sphere freed  from  dust.    The  presence  of  the  dust,  as  a  nucleus 
on  which  the  water  can  condense,  is  thus  an  essential  condition. 

203.  Water  Vapour. — The  presence  of  water  vapour 
has  already  been   discussed   in  the  chapter  on  hygrometry. 
Remembering  that  water  vapour  is  five-eighths  the  density  of 
dry  air  under  similar  conditions,  and  that  any  mixture  of  water 
vapour  and  air  is  less  dense  than  dry  air,  we  see  that  the 
heating  of  the  surface  of  the  earth,  which  by  conduction  heats 
the  lower  strata  of  air,  combined  with  the  result  of  the  water 
vapour  entering  the  air  on  account  of  evaporation,  will  be  to 
make  the  stratum  of  air  nearest  the  ground  less  dense  than 
that  above,  and  therefore  there  will  be  an  ascending  current  of 
atmospheric  air. 

204.  Pressure  and  Temperature,  and  Altitude. — 
The  atmospheric  pressure  is  due  to  the  column  of  air  above 
us,  and  it  follows  that,  generally,  as  we  ascend  the  pressure 
will  be  reduced.     As  the  greatest  action  of  the  sun's  heat  is  to 
raise  the  temperature  of  the  surface  of  the  earth,  the  lower 
strata  become  the  most  heated,  and  generally  the  nearer  the 
earth,  the   higher   the   temperature.     The    heated   air,    as    it 
ascends,    passes    into   a   region   of    lower   temperature,    and 
therefore  expands.     In  so  doing  it  does  work,  and  there  will 
be  a  consequent  fall  of  temperature.     If  the  air  were  perfectly 
dry,  it  has  been  computed  that  the  fall  would  be  i°  F.  for 
every  180  feet.    There  are,  however,  many  factors  to  consider. 


Applications —  Climate  3 1 7 

The  air  contains  moisture  that,  with  the  fall  of  temperature, 
condenses.  In  condensing,  heat  is  liberated,  and  the  fall  in 
temperature  is  retarded.  The  rate  of  fall  of  temperature  will 
vary  with  the  conditions,  but  an  approximate  rate  is  a  fall  of 
i°  F.  for  every  300  feet. 

205.  Condensation  of  Water  Vapour. — If  the  atmo- 
spheric air  be  cooled  sufficiently,  there  will  be  condensation 
of  the  moisture.  The  cooling  may  be  produced — 

(1)  As  we  have  seen  above,  by  the  ascent  of  moisture- 
laden  air  into  the  higher  altitudes. 

(2)  Warm  and  damp  currents  may  be  cooled  by  coming  in 
contact  with  cold  masses  of  mountains,  or,  in  certain  cases, 
of  earth. 

(3)  Warm  and  damp  currents  may  meet  colder  currents. 
In  all  cases  dust-particles  must  be  present. 
Condensation  and  the  production  of  rain  due  to  the  first 

cause  is  most  strikingly  seen  in  the  tropics.  The  rapid 
evaporation  of  water  and  the  heating  of  the  air  produce  a 
powerful  up-current ;  this  is  due  to  the  work  done  in  expansion 
and  also  to  the  fall  in  temperature  due  to  the  altitude.  The 
air  is  cooled,  and  part  of  the  moisture  condenses, — the  result  is 
almost  continuous  rain  in  the  tropical  regions  known  as  the  zone 
of  constant  precipitation ;  in  the  condensation,  however,  heat 
is  liberated,  and  excessive  condensation  is  retarded. 

To  the  second  cause  is  due  the  condensation  of  vapour 
as  rain  on  our  west  coasts,  and  in  certain  cases  to  the  ground 
fogs  near  cold  and  clayey  soils.  The  west  winds  bring  air 
laden  with  moisture ;  these  winds  impinge  upon  the  mountains 
on  the  west  coast,  and  the  stratum  at  the  base  is  in  part  cooled, 
and  rain  falls ;  but  the  air  is  forced  to  ascend,  therefore  the  first 
method  comes  into  play,  and  this,  added  to  the  coldness  of 
the  higher  parts  of  the  ranges,  produces  the  heavy  rainfall. 

The  third  mixture  is  not  so  effective  as  was  at  one  time 
supposed.  The  following  calculation  is  given  in  a  well-known 
text-book. 

"Suppose  a  cubic  metre  of  air  at  10°  C.  mixes  with  a 
cubic  metre  of  air  at  20°  CM  and  that  they  are  respectively 
saturated  with  aqueous  vapour.  It  is  easily  calculated  that 


318  Heat 

the  weight  of  water  contained  in  the  cubic  metre  of  air  at 
10°  C.  is  9*397  grams,  and  in  that  at  20°  C.  is  17*153  grams, 
or  26*559  grams  in  all.  When  mixed  they  produce  2  cubic 
metres  of  air  at  15°  C. ;  but  as  the  weight  of  water  required  to 
saturate  this  is  only  2  x  12-8=  25-6  grams,  the  excess,  0-95 
gram,  will  be  deposited  in  the  form  of  mist  or  cloud." 

But  0*95  gram,  condensing,  liberates  0*95  X  536  thermal 
units  =  509  thermal  units;  2  cubic  metres  of  moist  air  at  15° 
C.  weigh  about  2440  grams,  and  taking  the  specific  heat  as 
0-2375  (dry  air),  we  see  that  the  temperature  cannot  fall 
to  15°. 

Calculation  has  shown  that  the  rainfall  from  this  source 
must  be  small. 

206.  Dew. — Dew  is  the  condensed  vapour  formed  on 
terrestrial  objects  ;  the  vapour  is  condensed  on  the  object 
without  passing  through  the  form  of  cloud  and  then  falling  as 
rain.  The  explanation  of  this  phenomenon  is  due  to  Dr.  Wells. 

After  sunset,  if  the  sky  be  clear,  heat  is  radiated  from  the 
earth,  and  the  temperature  falls  ;  in  the  absence  of  much  water 
vapour  and  cloud,  the  temperature  of  parts  of  the  earth  falls 
below  the  dew-point  for  the  lowest  stratum  of  air,  and  dew  is 
deposited  on  these  objects. 

A  cloudy  night,  the  presence  of  much  aqueous  vapour, 
prevents  the  radiation,  and  dew  is  not  formed ;  it  is  also 
prevented  by  wind.  We  have  seen  that  rough  bodies  radiate 
heat  most  rapidly  ;  dew  is  hence  formed  on  grass  and  twigs 
when  none  is  formed  on  bright  objects — gravel  or  bright  metal 
tools.  The  formation  on  gravel  and  metals  is  further  prevented 
by  the  fact  that  they  are  good  conductors  of  heat,  and  the  loss 
due  to  radiation  is  made  up  by  conduction. 

The  slightest  covering  of  matting  or  similar  substance  over 
plants  is  sufficient  to  prevent  the  falling  of  temperature  due  to 
radiation  to  a  point  that  would  kill  the  plants. 

'If  the  temperature  be  below  freezing  point,  the  dew  is  at 
once  frozen,  and  appears  as  hoar-frost ;  if  the  dew  freezes  after 
it  is  deposited,  the  result  is  black  frost. 

Investigations  of  Mr.  Aitken  show  that  the  excessive  dew 
formed  on  some  plants  is  not  due  to  the  condensation  of 


Applications —  Climate  3 1 9 

vapour  of  the  air  alone,  but  is  also  due  to  vapour  given  out 
by  the  plant 

207.  Clouds  and  Rain. — When  moist  air  rises  to  high 
altitudes,  condensation  takes  place  when  the  temperature  of 
the  dew-point  is  reached;  the  small  particles  of  condensed 
water  form  clouds.     The  transparency  of  the  water  vapour  is 
replaced  by  a  certain  opacity  of  these  numerous  droplets,  and 
owing  to  the  number  of  the  reflecting  surfaces,  they  become 
visible.     These  droplets  are  always  falling  to  the  earth,  and 
in  falling  small  ones  coalesce  into  larger  ones;  if  they  pass 
through  an  unsaturated  stratum,  they  may  again  evaporate. 
Rain  is  therefore  always  descending  from  the  base  of  a  cloud, 
but  it  may  not  in  all  cases  reach  the  earth.     If  evaporation 
does  not  take  place  with  sufficient  rapidity,  the  drops  reach 
the  earth  as  rain.    In  condensation  heat  is  liberated,  explaining 
the  rise  in  temperature  that  accompanies  many  of  our  rains. 
A  gallon  of  rain,  for  example,  in  condensing,  will   liberate 
5360  units  of  heat;  as  the  specific  heat  of  air  is  low  (0-2375), 
it  is  at  once  evident  that  the  liberated  heat  will  have  a  marked 
effect  upon  the  temperature. 

208.  Snow   and   Hail. —  If   condensation   takes   place 
below  freezing  point,  the  water  crystallizes  as  ice,  the  general 
form  being  hexagonal ;  a  variety  of  shapes  are  formed,  and 
these  crystals  join  and  form  snowflakes.      Snow   sometimes 
falls  when  the  temperature  is  a  few  degrees  above  freezing 
point,  but  the  limit  is  soon  reached. 

The  formation  of  hail  has  not  been  satisfactorily  explained. 

209.  Mist  and  Fog.— The  name  mist  is  applied  to  the 
"cloud"  formed  near  the  surface  of  the  earth  by  the  cooling 
of  the  moist  air  below  the  dew-point ;  it  differs  from  fog  in 
that  the  particles  of  water  are  larger,  and  it  more  readily  wets 
objects  than  fog. 

The  fogs  in  large  towns  are  formed  by  the  vapour  con- 
densing on  the  small  particles  of  smoke;  the  soot-particles, 
covered  with  water  vapour,  remain  suspended  in  the  atmo- 
sphere, and  form  the  black  fogs. 

Warm  air  may  be  cooled  by  cold  currents  of  water  or  a 
cold  object,  and  a  fog  is  produced  ;  this  explains  the  dense 


32O  Heat 

fogs  formed  near  icebergs.  The  dense  fogs  near  Newfound-' 
land  are  due  to  the  cold  Arctic  current  meeting  air  warmed 
by  the  Gulf  Stream  (§  215). 

In  winter,  running  streams  (above  32°  F.)  are  frequently 
warmer  than  the  stratum  of  air  above  them ;  evaporation  takes 
place,  and  the  air  becomes  more  than  saturated  for  its  tempe- 
rature, and  a  fog  is  formed. 

On  clear  nights,  on  account  of  rapid  radiation,  the  tempera- 
ture of  the  ground  falls ;  the  stratum  of  air  nearest  the  ground 
is  cooled  below  its  dew-point,  and  dew  is  deposited.  This 
stratum  is  now  saturated,  and  colder  than  those  above  it ;  if  the 
configuration  of  the  ground  be  such  as  in  river  valleys  or  on 
low-lying  fields,  the  colder  air  above  flows  down  to  the  low- 
lying  parts,  and  by  mixture  produces  a  fog.  These  fogs  are 
thin  layers,  and  disappear  with  the  morning  sun,  the  tempera- 
ture soon  rising  sufficiently  high  to  enable  the  air  to  contain 
the  excess  of  water  vapour  that  constituted  the  fog. 

210.  Wind. — If  we  imagine  the  atmosphere  to  be  formed 
of  concentric  layers  around  the  earth,  the  lowest  layers  will  be 
the  most  dense,  and,  as  we  ascend,  the  density  of  the  layers 
will  diminish.     If  from  any  cause  the  lowest  layer  be  heated 
(this  is  done  in  nature  by  the  heated  earth),  that  layer  will 
expand  and  lift  the  layers  above, ^-a  blister,  so  to  speak,  will  be 
formed,  and  the  air  will  flow  away  on  all  sides  in  the  higher 
regions  of  the  atmosphere.     This  reduces  the  mass  of  air  over 
the  particular  spot,  and  consequently  there  will  be  a  reduction 
of  pressure  as  measured  by  the  barometer. 

At  the  surface  of  the  earth  the  heated  spot  will  become  a 
centre  of  low  pressure  compared  with  surrounding  places,  and, 
following  the  law  of  fluids,  air  will  flow  from  the  surrounding 
places  to  this  spot.  In  the  highest  layers  there  will  be  a 
current  away  from  the  spot,  and  thus  a  general  circulation 
will  ensue ;  this  is  assisted  from  the  fact  that  with  heat  evapo- 
ration takes  place,  and  the  mixture  of  heated  air  and  water 
vapour  formed  is  relatively  less  dense  than  the  air  immediately 
above  it,  and  thus  the  heated  moist  air  will  rise. 

211.  Land   and  Sea  Breezes. — Land  absorbs  radiant 
heat  from  the  sun  more  readily  than  the  sea ;  its  specific  heat 


Applications —  Climate  321 

is  also  lower  than  that  of  water;  the  water  also  reflects  the 
radiant  energy  more  readily  than  land,  and  its  incessant 
motion  presents  fresh  layers  to  be  heated,  and  this  again  pre- 
vents any  great  increase  in  its  temperature. 

During  the  day,  then,  the  land  will  become  relatively 
warmer  than  the  sea,  the  raising  of  the  layers  will  take  place 
over  the  land,  and  the  air  in  the  upper  layers  will  flow  seaward ; 
a  reduction  of  pressure  follows  over  the  land,  and  a  current  of 
air  (the  sea  breeze)  sets  in  from  the  sea.  As  soon  as  the  sun 
sets,  radiation  takes  place  rapidly  from  the  land;  the  temperature 
falls,  and  soon  falls  below  that  of  the  sea,  which  tends,  for  the 
reasons  given  above,  to  keep  fairly  constant.  It  follows  that 
the  low  pressure  will  now  be  over  the  sea,  and  a  land  breeze 
sets  in. 

212.  Effect  of  the  Earth's  Rotation  on  Winds. — 
The  direction  a  wind  or  current  takes  with  respect  to  the 
earth  will  be  modified  by  the  motion  of  the  earth  beneath  it 
Suppose  a  mass  of  air  in  the  northern  hemisphere,  at  rest  com- 
pared with  the  earth,  starts  with  a  velocity  of  say  20  feet  per 
second;  at  the  end  of  i  second  let  it  reach  a  point  that  is 
moving  eastward  with  a  velocity  of  2  feet  per  second  greater 
than  the  velocity  of  the  starting-point.  Then,  if  we  consider  a 
small  portion  of  the  earth's  surface — for  all  purposes  a  plane — 
and  neglect  any  effect  of  friction  between  the  air  and  the  earth, 
this  current  will  appear,  to  a  person  in  the  second  position,  to 
be  moving  from  the  north-east;  that  is,  if  the  wind  be  indicated 
on  a  map,  the  effect  of  rotation  has  been  to  turn  the  wind  to 
the  right  hand.  This  effect  will  be  seen  under  the  above  con- 
ditions in  whatever  direction  the  current  moves  in  the  northern 
hemisphere;  in  the  southern  hemisphere  the  effect  will  be 
to  twist  it  to  the  left  hand.  But  in  the  general  conditions  of 
atmospheric  circulation  we  have  a  mass  of  land  and  water 
surrounded  by  a  mass  of  air;  under  such  conditions,  if  we 
neglect  any  effect  of  friction,  a  particle  of  air  moving  towards 
the  equator  would  have  a  velocity  eastward,  compared  with 
that  of  the  earth,  that  would  vary  inversely  as  the  distance 
from  the  axis  of  rotation.  This  would,  in  the  northern  hemi- 
sphere, again  change  a  wind  that  originally  was  due  south  into 

Y 


322  Heat 

one  that  is  from  the  north-east ;  a  due  north  wind  would  be 
changed  into  one  that  was  from  the  south-west ;  and  the  change 
can  be  readily  applied  to  the  southern  hemisphere. 

In  reality,  the  lamina  of  air  above  the  earth  drags  upon  the 
surface  and  also  against  the  lamina  above  it,  so  that  a  wind 
starting  from  the  calms  of  Cancer  would,  as  it  journeyed 
towards  the  equator,  be  receiving  an  impetus  from  the  rotating 
mass  of  the  earth  that  would  tend  to  make  its  motion  less 
from  the  east  than  the  above  considerations  would  suggest ; 
and  the  wind  would  tend  to  have  the  same  absolute  velocity 
as  the  sea  and  land  beneath  it. 

213.  Atmospheric  Circulation.— Recorded  observa- 
tions show  that  a  belt  approximately  near  the  equator  is  a 
region  of  low  pressure,  that  a  belt  in  each  hemisphere  between 
20°  and  30°  latitude  is  a  region  of  high  pressure,  and  that  the 
poles  are  regions  of  low  pressure;  there  will  then  be  on  the 
surface  of  the  earth  a  wind  from  this  "  border  belt "  in  each 
hemisphere,  equator-wards  and  pole-wards,  and  by  §  212  the 
winds  to  the  equator  will  be  easterly  winds,  known  as  the 
trade  winds ;  the  winds  to  the  pole  will  be  westerly,  and  form 
the  north-westerly  and  south-westerly  winds  in  the  north  and 
south  hemispheres  respectively. 

These  surface  winds  necessitate  other  currents  in  the  upper 
regions  of  the  atmosphere  in  (generally  speaking)  opposite 
directions  to  the  surface  winds. 

The  following  explanation  of  atmospheric  circulation  was 
first  fully  described  by  the  late  Professor  James  Thomson,  in 
1857.  He  based  his  explanation  on  a  general  theory  pub- 
lished, in  1735,  by  Hadley.  The  whole  question  is  discussed 
in  the  Bakerian  Lecture  for  1892,  by  Professor  Thomson.1 

The  rising  of  the  heated  air  near  the  tropics  produces  a 
lowering  of  pressure  and  a  current,  which,  for  the  present,  we 
can  imagine  starting  from  the  calms  of  Cancer  (the  description 
that  follows  refers  to  the  northern  hemisphere  ;  the  student  can 
readily  adapt  the  explanation  to  the  southern  hemisphere) 
ensues.  This  wind,  by  the  last  section,  will  blow,  not  due 
south,  but  from  the  north-east ;  the  excess  of  westerly  direc- 
1  "  Philosophical  Transactions,"  1892. 


Applications —  Climate 


323 


tion  over  the  earth  will,  however,  by  friction  of  the  earth,  be 
diminished,  so  that  as  it  approaches  the  equator  it  becomes 
less  and  less  an  easterly  wind,  and  it  ascends  with  an  absolute 
easterly  velocity  but  slightly  less  than  the  rotational  velocity 
of  the  earth ;  that  is,  its  absolute  easterly  direction  is  approxi- 
mately 1000  miles  an  hour,  while  its  relative  westerly  direction 
compared  with  the  earth  is  approaching  zero.  We  can  with 
convenience  adopt  the  convention  of  Professor  Thomson, 


FIG.  135. 

and  call  air  that  has  no  eastward  or  westward  motion  relative 
to  the  earth's  surface  as  having  par,  or  being  at  par  of  revo- 
lutional  velocity,  and  likewise,  to  use  the  designation  over  par 
of  revolutional  velocity  to  signify  eastward  relative  motion,  and 
under  par  to  signify  westward  relative  motion. 

The  air,  then,  rises  at  the  belt  near  the  equator  with  par 
revolutional  velocity,  and  in  the  upper  regions  flows  north  and 


324  Heat 

south.  If  we  follow  the  northern  current,  we  see  that,  as  it 
advances  to  the  north,  it  has  a  greater  eastward  velocity  than 
the  earth  beneath  it ;  it  cannot  lose  its  revolutional  momentum 
by  reason  of  any  fluid  above  it,  and  will  be  but  slightly  affected 
by  the  current  beneath  it;  that  is,  it  is  moving  north  with  over- 
par  revolutional  velocity.  It  gradually  cools  as  it  moves 
towards  the  higher  latitudes,  it  is  also  deprived  of  part  of  its 
water  vapour,  and  becomes  more  dense ;  this  north-moving 
sheet  of  air  engirdling  the  earth  will  descend  in  the  middle 
and  higher  latitudes,  and  should  then  return  towards  the 
equator.  Remembering  that  its  revolutional  velocity  is  over 
par,  it  should  return  as  a  wind  from  the  north-west.  As  a 
result  of  observation,  the  general  wind  in  middle  latitudes  is 
from  the  south-east.  To  explain  this,  Professor  Thomson 
states  that  the  greatest  amount  of  air  does  return  as  a  current 
from  the  north-west,  and  that  this  constitutes  the  main  return 
current.  This  return  current,  when  it  turns  south,  is  at  over- 
par  revolutional  velocity,  and  there  will  be  a  sheet  of  air 
engirdling  the  earth  beneath  the  upper  current  (also  at  over 
par),  and  this  sheet  of  air  at  over-par  revolutional  velocity 
induces,  upon  a  comparatively  thin  lamina  of  air  next  to  the 
earth,  a  wind  towards  the  north-east,  that  is,  pole-wards,  with 
over-par  revolutional  velocity  (Fig.  135,  after  Thomson). 
He  illustrates  this  by  the  following  experiment : — 
"  If  a  shallow  circular  vessel  with  flat  bottom  be  filled  to  a 
moderate  depth  with  water,  and  if  a  few  small  objects,  very 
little  heavier  than  water,  and  suitable  for  indicating  to  the  eye 
the  motions  of  the  water  in  the  bottom,  be  put  in,  and  if  the 
water  be  set  to  revolve  by  being  stirred  round,  then,  on  the 
process  of  stirring  being  terminated,  and  the  water  being  left 
to  itself,  the  small  particles  in  the  bottom  will  be  seen  to 
collect  in  the  centre.  They  are  evidently  carried  there  by  a 
current  determined  towards  the  centre  along  the  bottom  in 
consequence  of  the  centrifugal  force  of  the  lowest  stratum  of 
the  water  being  diminished  in  reference  to  the  strata  above, 
through  a  diminution  of  velocity  of  rotation  in  the  lower 
stratum  by  friction  on  the  bottom.  The  particles,  being 
heavier  than  the  water,  must  in  respect  of  their  density  have 


Applications — Climate  325 

more  centrifugal  force  than  the  water  immediately  in  contact 
with  them,  and  must  therefore,  in  this  respect,  have  a  tendency 
to  fly  outwards  from  the  centre ;  but  the  flow  of  water  towards 
the  centre  overcomes  this  tendency,  and  carries  them  inwards, 
and  thus  is  the  flow  of  water  towards  the  centre  in  the  stratum 
in  contact  with  the  bottom  palpably  manifested." 

The  indraught  to  the  centre  of  the  vessel  is  similarly 
observable  when  the  vessel  itself  is  rotating,  provided  that  its 
velocity  of  rotation  is  less  than  that  of  the  fluid. 

Applying  this  to  atmospheric  circulation,  we  have  in  middle 
latitudes  and  towards  the  poles  an  upper  current  from  the 
equator  to  the  poles  with  over-par  revolutional  velocity,  below 
this  the  main  return  current  to  the  equator  also  with  over-par 
velocity,  and  the  lamina  in  contact  with  the  earth  still  with 
over-par  velocity,  but  with  an  indraught  towards  the  poles ;  it 
is  this  lowest  stratum  that  is  observed,  and  forms  the  general 
south-west  winds  of  the  middle  and  northern  latitudes  of  the 
northern  hemisphere. 

There  will  be,  then,  three  strata  (see  Fig.  135)  of  air  in  the 
higher  latitudes.  If  we  follow  the  middle  main  return  current 
above  the  lowest  strata,  we  shall  expect  that  it  will  gradually 
lose  part  of  its  over-par  velocity,  seeing  that  it  has  to  drag 
forward  the  lower  layer  ;  about  the  calms  of  Cancer  it  has  lost 
so  much  of  its  over-par  revolutional  velocity  that  it  is  no 
longer  able  to  have  an  effective  result  on  the  lowest  lamina, 
and  therefore  the  indraught  northward  ceases.  The  return 
current  now  forms  the  lowest  stratum,  and  comes  under  the 
influence  of  the  indraught  towards  the  equator,  and  the  cir- 
culation is  complete,  the  north-east  trade  wind  again  being 
produced. 

The  lamina  below  the  return  main  current  as  it  flows  north 
is  gradually  deprived  of  its  over-par  velocity,  with  the  result 
that  over  the  pole  there  will  be  impounded  a  great  mass  of 
air;  that,  it  .is  suggested,  lies  there,  and  is  only  brought  into 
circulation  by  being  dragged  away  by  the  great  upper  current 
that  is  possessed  of  great  revolutional  motion. '  It  will  be  seen 
that  the  mid  return  current  is  also  the  return  current  for  the 
lowest  lamina. 


326  Heat 

The  general  direction  will  be  modified  and  altered  by  the 
land  and  changes  in  climatic  conditions.  For  the  particular 
discussion,  as  well  as  reference  to  particular  winds,  the  student 
may  consult  the  work  in  this  series  dealing  with  Physiography. 

214.  Water    Circulation. — Ocean    Currents.— The 
result  of  deep-sea  soundings  has  shown  that  the  ocean,  save 
a  comparatively  thin  upper  stratum  (that  varies  in  depth  with 
position  and   time  of  the   year)  is  composed  of  water  but 
slightly  above  freezing  point.     This  mass  of  cold  water  is  pro- 
bably due  to  the  action  of  the  ice  and  cold  of  the  polar  regions. 

In  the  equatorial  regions  the  sun  heats  the  surface  of  the 
sea;  expansion  takes  place,  and  a  circulation  ensues  that 
would  be  primarily  a  surface  current  from  equator  to  pole,  and 
a  current  at  the  bottom  of  the  ocean  from  pole  to  equator. 
The  currents  are  modified  by  the  rotation  of  the  earth  (com- 
pare pp.  321,  322)  and  the  direction  of  the  winds. 

215.  The  Gulf  Stream. — As  a  result  of  the  trade  winds 
and  the  rotation   of  the  earth,  a  current  sweeps  across  the 
South  Atlantic ;  this  current  is  divided  into  two  portions  by 
Cape  St.  Roque,  the  northern  portion  flowing  along  the  coast 
of  South  America  into  the  Caribbean  Sea,  and  is  reinforced 
at  the  islands  by  a  current  from  Cape  Verde  Islands. 

The  large  mass  of  warm '  water  from  tropical  regions 
crowded  into  the  Caribbean  Sea  is  forced  through  the  Straits 
of  Florida,  and  is  reinforced  by  other  branches  of  the  current, 
that,  divided  by  the  islands  from  that  which  entered  the 
Caribbean  Sea,  has  swept  past  Cuba  and  St.  Domingo,  and 
joins  the  current  generally  called  the  Gulf  Stream.  The 
stream  passing  through  the  straits  is  40  miles  wide,  3000  feet 
deep,  and  has  a  velocity  at  the  middle  of  about  5  miles  an 
hour.  Sweeping  north,  it  grows  wider  and  shallower;  near 
Newfoundland  its  width  is  320  miles,  in  the  middle  Atlantic 
800  miles.  The  prevailing  south-west  winds  now  influence  it. 
In  mid- Atlantic  it  divides;  one  branch,  either  a  continuation  of 
the  Gulf  Stream,  or  due  to  the  action  of  the  prevailing  winds,  a 
new  current  formed  out  of  the  water  brought  by  the  Gulf 
Stream,  flows  to  the  British  Isles,  Iceland,  and  Norway;  the 
other,  affected  by  the  earth's  rotation,  turns  to  the  right,  and 


Applications —  Climate  327 

passes  round  by  Spain  and  the  north  coast  of  Africa,  and  joins 
the  main  equatorial  Atlantic  current;  the  main  currents  sur- 
round a  central  mass  of  water  forming  the  Sargasso  Sea. 

It  should  be  noticed  that  a  marked  effect  of  the  Gulf 
Stream  on  these  isles,  as  regards  heat,  is  due,  not  directly  to  the 
current  of  water,  but  to  the  fact  that  the  prevailing  south-west 
winds  flow  over  it,  become  charged  with  moisture,  and,  on 
condensation  taking  place,  heat  is  liberated  (§  207). 

216.  Electric  Pyrometer. — In  §  152  it  was  seen  that  if 
one  junction  of  two  metals  be  kept  at  a  higher  temperature 
than  the  other  junctions,  a' current  is  produced  that  varies  with 
the  difference  in  temperature  of  the  junctions.  Instruments 
have  been  made  based  on  this  principle  for  the  measurement 
of  high  temperatures. 

Two  wires— for  temperatures  up  to  about  225°  C.,  copper 
and  iron  may  be  used — are  joined  at  one  junction  by  wrapping 
platinum  wire  round  them  for  a  short  distance, — they  are 
otherwise  kept  separate  throughout  their  lengths.  The  wires 
are  placed  in  a  porcelain  tube ;  the  other  ends  are  soldered 
or  joined  to  thick  copper  wires  that  lead  to  a  galvanometer. 
The  soldered  or  joined  ends  are  kept  at  a  constant  temperature 
by  immersing  them  in  water  at  the  temperature  of  the  room, 
or  by  placing  them  in  melting  ice. 

The  ends  joined  by  platinum  are  now  subjected  to  known 
temperatures  by  immersing  them  in  solutions  or  in  metals  at 
their  boiling  points,  and  the  deflection  of  the  galvanometer  is 
observed  for  each  of  these  known  temperatures.  The  deflec- 
tion is  kept  small  by  introducing  suitable  resistances  into  the 
circuit ;  from  the  observed  deflections  estimates  are  made  as 
to  the  temperature  indicated  by  any  difference  between  the 
observed  deflections. 

By  placing  the  ends  of  the  wires  in  any  enclosure,  it  is  thus 
possible  to  estimate  the  temperature. 

At  276°  C.  in  the  case  of  iron  and  copper  the  neutral 
point  is  reached,  and,  if  the  temperature  be  raised  higher,  the 
current  is  reversed;  before  this  temperature  is  reached  the 
deflections  produced  are  small  for  a  slight  change  of  tem- 
perature. 


328 


Heat 


In  Becquerel's  pyrometer,  platinum  and  palladium  wires  are 
used;  these  are  less  fusible  than  copper  and  iron,  and,  proceed- 
ing as  above,  a  scale  can  be  formed  for  these  wires. 

217.  Siemens's  Electrical  Pyrometer. — The  current 
from  a  battery,  B,  flows  along  the  wire  BAG;  at  C  it  divides. 
One  portion  passes  along  the  coiled  platinum  wire  D  to  E,  and 
enters  one  coil  of  the  differential  galvanometer  G  by  a,  leaves 
it  by  b,  and,  passing  along  the  wire  H  B,  returns  to  the  battery 
B.  The  other  portion  of  the  current  flows  from  C  along  the 
wire  C  J  to  the  resistance  box  R ;  it  passes  through  the 
resistance  box,  and  then  flows  through  the  other  coil  of 


FIG.  136. 

the  differential  galvanometer  G ;  it  enters  this  coil  at  c,  and 
leaves  it  at  d\  it  then  flows  to  H,  and  passes  along  H  B  to 
the  battery. 

If  the  resistance  of  the  branch  C  D  E  a  b  H  be  equal  to 
the  resistance  of  the  branch  C  J  R  c  d  H,  the  current  will  divide 


Applications — Climate  329 

into  equal  portions  at  C,  and  there  will  be  no  deflection  of  the 
needle  of  the  galvanometer. 

C  D  is  a  coil  of  platinum  wire  wrapped  round  a  fire-clay 
cylindrical  cell ;  the  wire  is  surrounded  by  asbestos,  that  is 
kept  in  position  by  platinum  foil ;  the  protected  coil  is  placed 
inside  a  long  wrought-iron  tube,  M  N,  and  is  kept  in  position 
by  a  packing  of  lime ;  the  packing  of  lime  also  serves  to  keep 
the  wires  C  A,  C  J,  and  D  E  in  position. 

In  graduating  the  instrument,  the  end  of  the  tube  con- 
taining C  D  is  placed  in  baths  whose  temperatures  are  known, — 
for  example,  boiling  water,  boiling  zinc ;  the  resistance  of  C  D 
is  increased,  and  the  resistance  of  R  is  adjusted  so  that  there 
is  no  deflection  in  the  galvanometer.  The  results  are  plotted 
so  that  the  resistances  in  ohms  form  the  ordinates,  and  the 
temperatures  the  abscissae;  it  is  found  that  practically  the 
curve  formed  is  a  straight  line. 

In  order  to  determine  any  unknown  temperature,  the  closed 
end  of  the  tube  is  subjected  to  the  unknown  temperature,  and 
the  resistance  necessary  to  prevent  any  displacement  of  the 
galvanometer  needle  is  read.  From  the  plotted  curve  the 
unknown  temperature  is  readily  deduced. 

By  means  of  pyrometers  of  this  description  it  is  possible 
to  determine  temperatures  up  to  1200°  C. 

They  are  much  more  effective  instruments  than  the  water 
pyrometer  described  in  §  690. 

EXAMPLES.    XIII. 

1.  State  clearly  how  you  suppose  winds  to  be  produced. 

2.  Wherein  does  the  manner  in  which  heat  is  usually  diffused  through 
liquids  and  gases  differ  from  the  mode  of  its  diffusion  through  solids  ? 

3.  Explain  the  formation  of  dew,  hoar-frost,  and  black  frost. 

4.  How  do  you  account  for  the  fact  that  a  cloud  is  sometimes  formed 
by  the  mixture  of  two  quantities  of  air  at  different  temperatures,  although 
neither  quantity  is  quite  saturated  before  the  mixture  ? 

5.  A  building  is  heated  by  hot- water  pipes.    How  does  the  heat  get 
from  the  furnace  of  the  boiler  to  a  person  in  the  building  ?     What  would 
be  the  effects  on  the  temperature  of  the  more  distant  parts  of  the  building 
of  coating  the  pipes  near  the  boiler  (a)  with  woollen  felt,  (l>)  with  dull  black 
lead.     (London  Matric.) 


33O  Heat 

6.  Explain  the  construction  of  Siemens's  electrical  pyrometer.     How 
would  you  proceed  to  graduate  such  an  instrument  ? 

7.  Explain  Professor  J.  Thomson's  theory  of  atmospheric  circulation. 
Account  carefully  for  the  prevailing  winds  in  our  latitude. 

8.  Give  an  explanation  of  the  Gulf  Stream.     How  does  it  affect  the 
climate  of  Western  Europe  ? 

9.  If  I  Ib.  of  coal  in  burning  will  raise    14,000  Ibs.  of  water  one 
degree  Fahrenheit,  and  if  an  engine  can  convert  one-fifth  of  the  heat  given 
it  to  useful  work,  find  how  much  coal  must  be  burnt  in  raising  I  ton  of 
coal  up  a  pit-shaft  600  yards  deep.     (London  Inter.  Sc.) 

10.  Explain  the  cause  of  land  and  sea  breezes. 

11.  The  specific  heat  of  zinc  is  O'OpS,  and  280  grams  of  zinc  are  raised 
to  the  temperature  of  97°  C.,  and  immersed  in  180  grams  of  water  at  14°  C., 
contained  in  a  copper  calorimeter  weighing  96  grams,  the  specific  heat  of 
copper  being  0*095  :  what  will  be  the  temperature  of  the  mixture,  sup- 
posing that  there  is  no  exchange  of  heat  except  among   the  substances 
mentioned  ?    What  is  the  water  equivalent  of  the  calorimeter  employed  ? 
(London  Inter.  Sc.) 


SCIENCE   AND   ART   PAPERS. 

Second   Stage,  or   Advanced   Examination. 

MAY,  1892. 

1.  What  is  meant  by  the  hygrometric  state  of  the  air?  and  state  how 
you  would  determine  it  by  a  condensation  hygrometer. 

2.  Describe  a  method  of  measuring  the  specific  heat  of  a  gas  at  con- 
stant pressure. 

3.  A  piece  of  iron  weighing  16  grams  is  dropped  at  a  temperature  of 
112-5°  C.  into  a  cavity  in  a  block  of  ice,  of  which  it  melts  2*5  grams  : 
if  the  latent  heat  of  ice  is  So,  find  the  specific  heat  of  iron. 

4.  Two  liquids,  A  and  B,  are  introduced  into  two  barometer  tubes,  the 
temperature  of  each  being  the  same.     It  is  noticed  (i)  that  in  both  cases 
a  little  of  the  liquid  does  not  evaporate  ;  (2)  that  the  mercury  in  the  tube 
containing  A  is  more  depressed  than  that  in  the  tube  into  which  B  was 
introduced.     Which  liquid  would  you  expect  to  have  the  higher  boiling 
point  ?     Give  reasons  for  your  answer. 

5.  If  the  coefficients  of  cubical  expansion  of  glass  and  mercury  are 
O'c.00025  and  o  00018  respectively,  what  fraction  of  the  whole  volume  of 


Science  and  Art  Papers  331 

a  glass  vessel  should  be  filled  wilh  mercury  in  order  that  the  volume  of 
the  empty  part  should  remain  constant  when  the  glass  and  mercury  are 
heated  to  the  same  temperature  ? 

6.  Describe  the  phenomena  observed  during  the  fusion  or  solidification 
of  an  alloy  of  two  metals,  and  give  some  explanation  of  the  phenomena. 

7.  Describe  the  apparatus  used  in  the  liquefaction  of  oxygen. 

8.  What  do  you  understand  by  the  first  law  of  thermo-dynamics  ?  and 
how  has  its  truth  been  established  ? 

9.  Distinguish  between  a  gas  and  a  vapour.     How  would  you  show 
that  the  pressure  of  a  mixture  of  gases  and  vapours  between  which  there 
is  no  chemical  action,  is  equal  to  the  sum  of  the  pressures  which  each 
would  severally  exert  if  alone  present  ? 


MAY,  1893. 

1.  Define  the  terms  "  energy  "  and  "  work,"  and  explain  ho\v,  first,  the 
kinetic,  secondly,  the  potential  energy  of  a  falling  body  is  measured. 

2.  Describe  the  constant  volume  air-thermometer.      How  would  you 
use  it  to  find  the  absolute  zero  of  the  air-thermometer? 

3.  Twenty-five  grams  of  water  at  15°  C.  are  put  into  the  tube  of  a 
Bunsen  ice  calorimeter,  and  it  is  observed  that  the  mercury  moves  through 
29  centimetres.     Fifteen  grams  of  a  metal  at  100°  C.  are  then  placed  in 
the  water,  and  the  mercury  moves  through  12   centimetres.      Find  the 
specific  heat  of  the  metal. 

4.  Define  the  dew-point,  and  explain  how  to  find  the  mass  of  aqueous 
vapour  present  in  a  given  volume  of  air. 

5.  Define  the  critical  point  of  a  fluid.     Give  sketches  of  and  point  out 
the  differences  between  the  forms  of  the  isothermals  of  carbonic  acid  above 
and  below  its  critical  point. 

6.  Enunciate  the  axiom  on  which  the  second  law  of  thermo-dynamics 
rests,  and  show  how   to  deduce   from   it   a  proof  of  the   fact   that   the 
efficiency  of  a  simple  reversible  cycle  is  a  maximum. 

7-  Describe  and  explain  the  spheroidal  state  of  a  liquid. 

8.  The  latent  heat  of  steam  at  100°  C.  is  536.     If  a  kilogram  of  water, 
"when  converted  into  saturated  steam  at  atmospheric   pressure,  occupies 
1*651  cubic  metre,  calculate  the  amount  of  heat  spent  in  internal  work 
during  the  conversion  of  water  at  100°  C.  into  steam  at  the  same  tempera- 
ture. 

9.  Describe  some  form  of  electrical  pyrometer. 


ANSWERS   TO    EXAMPLES 

I.  (4)  F.°,  212,  32,  -459'4,  -36'4,  392  ;  R.°,  80,  o,  -218-4,  -3°'4> 

160.     (5)  100,  o,  -35'5»  -273*3-     (7)  6'6.     (8)  49. 

II.  (i)  10-0085  feet,   10-017  feet,  10-019  feet.    (2)  Density  at  212°: 

density  at  32°  ::  10,000  :  10,024.  (3)  526*02  cubic  cm. 
(4)  0-42  inch,  say  J  inch.  (5)  At  25°  C.  brass  :  iron  : : 
1,000,000  I  1,000,035 ;  at  — 10°  C.  brass  :  iron  : :  1,000,000  : 
999,912.  (6)  0-000000175;  with  correction  (see  Errata), 
0-0000175.  (7)  0-000043,  0-000014. 

III.  (2)  i -54  cubic  cm.     (6)  0*00018. 

IV.  (i)  1-464  cubic  feet.     (3)  718-98  grams.     (5)  0*0002.     (6)  4786 

grams.  (7)  1307*6  inches  of  mercury  =  89  atmospheres 
nearly.  (8)  6432*8  mm.  of  mercury  =  8-5  atmospheres 
nearly,  (u)  0-001294  grams. 

V.  (i)  172  Ib.  nearly.  (2)  551-3°  C.  (3)0-091.  (4)0*639.  (5) 
0-0000188.  (6)  320°  C.  (7)  0-0329.  (8)  0*932.  (10) 
0-0819.  (u)  877.  (12)  30700  (air),  30545  (hydrogen). 
VI.  (2)  158*5.  (3)  (a)  8°;  (t)  46*8°;  (c)  7*7°;  (d)  24*7°  (s.  h.  of 
phosphorus  =  0-147).  (4)  (a)  277*56  t.  u.  =  385808*4  foot- 
pounds; (b)  263*94  t.  u.  =36,687,660  foot-pounds.  (5) 
o-iio.  (6)  5*66 Ibs.  ( 10)  740*34  (see  §  107).  (n)o*95°C. 
VII.  (3)  966-6.  (4)  44'4°  C.  (5)  195  :  "2.  (7)  (a)  2i*i°  C. ;  (b) 
25*6°  C.  (8)  624*8,  637.  (9)  4-8  Ibs.  (10)  536*2  (in  ques- 
tion, for  3*82  read  3*32).  (i  i)  263 -3  cubic  cm.  (13)  228°  F., 
250°  F.,  267°  F. 

VIII.     (12)  (i)  16-39;  (2)  16-06.     (13)  4-27  (air=  i). 
IX.     (2)  0*68.     (3)  0*75.     (4)  6-08  grams.     (6)  0*74.     (7)  0*102  gram, 

79  cubic  cm.     (8)  loo  :  80. 

X.     (3)  30*1  thermal  units.     (4)  16848.     (6)  0-153. 
XI.     (4)  4  :  i.     (15)  0-001417. 

XII.     (10)  V«-     (13)  ifS,  100,361  foot-pounds.     (14)  16-9°  C. 
XIII.     (9)  1*87 Ib.     (IT)  24-5°  C.  ;  9*12  grams. 
Science  and  Art — 

1892.  (3)  o*i.     (5)  & 

1893.  (3)  53g.      (8)  Internal   work  :  external  work  ::  12*33  •  J>   •"•  tne 

amount  of  heat  spent  in  internal  work  is  }§§§  of  the  whole 
=  495. 790  t.  u. 


INDEX 


ABSOLUTE  temperature,  air-ther- 
mometer, 68 

Absolute  temperature,  thermo- 
dynamic  scale,  301 

Absolute  zero,  63 

Absorption,     selective,     242,    252, 

259 

Absorption  and  emission,  252 
Adiabatic    changes,    283;    curves, 

283,  309 
Air,  cooling  by  ascent,  316  ;  density 

of,  70 ;  liquid,   183  ;  moisture  in 

the,     193  ;     thermometers,     69  ; 

weight  of,  204 
Alcohol  thermometer,  13 
Alloys,  fusion  of,  189 
Amagat's  researches,  68 
Andrews's  critical  point,  177 
Apjohn's  formulae,  201 
Apparent  and  absolute  expansion, 

34 

Atmosphere,  pressure  of,  54 
Atmospheric  circulation,  322 
Atomic  weights,  98 
Avogadro's  law,  190 

BAROMETRIC  readings,  52 

Black,  on  latent  heat  of  fusion, 
129;  on  latent  heat  of  vaporiza- 
tion, 157 

Boiling,  151 

Boiling  point,  7  ;  effect  of  pressure, 
17,  153;  to  determine,  152 

Boiling  points,  table  of,  152 

Bolometer,  251 

Bottomley's  ice  experiment,  137 

Boutigny  on  spheroidal  state,  184 

Boyle's  law,  57,  68,  74 

Breezes,  land  and  sea,  320 

Bunsen's  calorimeter,  89 


CAGNIARD-LATOUR  on  liquefaction, 
177 

Cailletet,  on  Boyle's  law  at  high 
pressures,  61  ;  liquefaction  of 
gases,  1 80 

Calibration,  10 

Caloric  theory,  105 

Calorimeter,  85 

Calorimeter,  Bunsen's,  89 ;  Lavoi- 
sier and  Laplace's,  88 

Capacity,  thermal,  83 

Carbonic  acid,  liquid,  178:  solid, 
182 

Carnot's  cycle,  292 

Carnot's  engine,  292 

Carre's  ice-machine,  163 

Centigrade  scale,  9 

Charles's  law,  68,  74 

Chemical  hygrometer,  204 

Chimney,  draught  of,  313 

Climate,  315-327 

Clouds,  319 

Coefficient  of  expansion,  19,  29 

Cold  of  evaporation,  161 

Cold,  radiation  of,  240,  261 

Compensated  pendulums,  28,  51 

Condensation,  144,  170 

Conduction  of  heat,  208 ;  table, 
212 ;  gases,  223 ;  liquids,  221  ; 
solids,  207 

Conductivity,  212  ;  absolute,  213  ; 
dimensions  of,  218 ;  of  crystals, 
220 ;  of  gases,  223  ;  of  liquids, 

221 

Convection,  207 ;    of  air,   207 ;   of 

water,  207 
Cooling,  law  of,  254,  266  ;  method 

of,  91 ;  by  ascent  of  air,  316 
Critical  temperature,  180 
Crookes's  radiometer,  262 


334 


Index 


Cryohydrates,  135 
Cryophorus,  164 
Currents,  air,  322 ;  ocean,  326 
Cycle  of  operations,  291 
Cycle,  reversible,  298 

DALTON'S  law  of  mixed  vapours, 

150 

Daniell's  hygrometer,  197 
Davy  and  Geordy  lamps,  224 
Davy  on  caloric,  105 
Density  and  temperature,    31  ;    of 

air,  70 ;  of  gases,  70 ;  table  of, 

72,  74 

De  Saussure's  hygroscope,  194 
Despretz,  conduction,  2IO 
Dew,  197-204,  318 
Dew-point,  197-204,  318 
Dewar  on  liquid  oxygen  and  air, 


183 
>iathe 


Diathermancy,  245 

Differential  thermometer,  16,  226 

Diffusion  of  heat,  241 

Diffusivity,  216 

Dimensions  of  quantities,  117,  260 

Dines's  hygrometer,  198 

Distillation,  166 

Distillation,  fractional,  167 

Drion's  experiment,  177 

Dulong  and  Petit  on  absolute  ex- 
pansion, 39  ;  on  atomic  heat,  98  ; 
law  of  cooling,  256 

Dust,  action  of,  316 

EFFICIENCY  of  heat-engines,  298 

Emission,  242 

Emissivity,  259 

Energy,  115;  kinetic,  115;  poten- 
tial, 115 

Engines,  reversible,  299 

Entropy,  309 

Evaporation,  165 

Evaporation,  cold  due  to,  161 

Exchange,  theory  of,  261 

Expansion,  3,  19 ;  apparent  and 
real,  of  liquids,  34  ;  coefficient  of, 
19,  29 ;  coefficient  table,  26  ; 
cubic,  29 ;  force  of,  27  ;  on  freez- 
ing, 136  ;  of  gases,  65-68  ;  linear, 
19-25  ;  of  liquids,  4,  51  ;  of  mer- 
cury, 37  ;  of  solids,  3,  49  ;  square, 
29  ;  true  coefficient  of,  45 

FAHRENHEIT'S  scale,  8 


Faraday,  liquefaction  of  gases,  175 

Faraday  on  regelation,  137 

Fireplaces,  313 

Fire-syringe,  109 

Fixed  points,  63  ;  variation  of,  II 

Fog,  319 

Forbes  on  conductivity,  214 

Freezing     by     evaporation,      163  ; 

mercury,  164;  mixtures,  133 
Freezing    point,    6  ;     lowered    by 

pressure,  137,  138,  305,  307 
Friction,  heat  of,  121 
Frost,  hoar,  318;  black,  318 
Fusion,    128 ;    latent  heat  of,    88, 

128;    laws   of,    128;    of  alloys, 

139 

GASES,      conductivity     of,      223  ; 

densities   of,    70 ;    expansion   of, 

65-68 ;    kinetic   theory  of,    124 ; 

liquefaction  of,  170,  173;  specific 

heats  of,  95 

Gay  Lussac  on  expansion  of  air,  61 
Gay    Lussac    on    vapour-densities, 

1 86 

Glaciers,  138 

Glaisher's  factors  and  tables,  202 
Gravesande's  experiment,  4 
Gridiron  pendulum,  28 
Gulf  Stream,  326 
Guthrie  on  cryohydrates,  135 

HAIL,  319 

Heat,  and  work,  109 ;  as  a  quantity, 

79 ;    mechanical    equivalent    of, 

121;  molecular  theory  of,    124; 

nature  of,  103  ;  units,  79 
Heating    by   hot   water,    314 ;    by 

steam,  315 
!   Heights  measured  by  boiling  point, 

J55 

Hoar-frost,  318 

|   Hoffmann,  vapour-density,  186 
|    Hope's  experiment,  46 
j    Humidity,  195 
Hydrogen,  mechanical  equivalent  of 

heat  deduced  from,  277 
!   Hygrometers,       chemical,        204  ; 
Daniell's,     197  ;     Dines's,     198  ; 
Mason's,  200  ;  Regnault's,  199 
j    Hygroscopes,  194 
Hypsometer,  155 

!   ICE  calorimeter.  88 


Index 


335 


Ice-regelation,  137 
Indicator  diagram,  288 
Ingenhouzs's  experiments,  209 
Intensity  of  heat,  237 
Inverse  squares,  law  of,  235 
Isothermal  lines,  58,  170 

JOLY'S  air- thermometer,  68 
Joly  on  specific  heat,  97 
Joule's  equivalent,  121 
Joule  on  internal  work,  270,  272 

LAMPBLACK,  as  absorber,  243  ;  as 
radiator,  244;  as  reflector,  241 

Land  and  sea  breezes,  320 

Langley  on  radiant  heat,  251 

Langley's  bolometer,  251 

Laplace  and  Lavoisier  on  expan- 
sion, 20 ;  ice  calorimeter  of,  88 

Latent  heat  of  fusion,  88,  128  ;  of 
vaporization,  156;  of  water,  158 

Leslie's  differential  thermometer, 
1 6,  226 ;  experiment  on  boiling, 

!53 

Liquefaction  of  gases,  170,  173 
Liquid  and  gaseous  states,  173 

MASON'S  hygrometer,  200 
Maximum   and    minimum   thermo- 
meters, 13 

Maximum  density  of  water,  46 
Mayer  on  mechanical  equivalent  of 

heat,  268 
Mechanical  equivalent  of  heat,  121, 

277 

Melloni's  experiments,  245,  246 
Melting    point,    127;    lowered    by   j 

pressure,  138 

Mendeleeffon  density  of  water,  50 
Mercury  as  thermometric  substance, 

16 ;   absolute  expansion  of,   37  ;    I 

freezing  of,  164 
Meyer    (Victor),    vapour-densities, 

189 

Mist,  319 

Mixtures  of  gases  and  vapours,  150 
Mixtures,  freezing,  133 
Mixtures,  method  of  specific  heat,  84 
Moist  air,  193,  317 
Mousson's  experiment,  138 

NEWTON,  law  of  cooling,  254 
Newton,  velocity  of  sound,  281 


Nobili's  thermo-pile,  226 
OCEAN  currents,  326 

PAPIN'S  digester,  154 
Pendulum,  compensated,  28 
Pendulum,  Graham's,  51 
Permanent  gases,  176 
Pictet,  liquefaction  of  gases,  181 
Pressure  of  vapours,  146 
Prevost,  theory  of  exchanges,  261 
Pyrheliometer,  265 
Pyrometer,  electric,  327  ;  Siemens's 
electrical,  328 ;  water,  100 

QUANTITY,  heat  as  a,  79 

RADIANT  heat,  229  ;  and  light,  232 

Radiation,  226 

Radiometer,  262 

Rain,  319 

Real  and  apparent  expansion,  34 

Reaumur's  scale,  9 

Reflecting  power,  241 

Reflection  of  heat,  238 

Refraction  of  radiant  heat,  247 

Regelation,  137 

Regnault,      hygrometer,     199 ;     on 

absolute    expansion   of  mercury, 

41  ;    on    densities   of  gases,   70 ; 

on  expansion  of  gases,  65-68  ;  on 

vapour-pressures,  147 
Reversible    engine,  299;   efficiency 

of,  299 
Roy  and    Ramsden  on  expansion, 

21 

Rumford  on  heat  of  friction,  106 
Rutherford's  thermometers,  14 

SALINE  solutions,  155 

Saturated  vapour,  145 

Saturation,  134 

Scales,  thermometric,  8 

Selection,  absorption,  and  emis- 
sion, 252 

Six's  thermometer,  14 

Snow,  319 

Solar  radiation,  264 

Solidification,  134;  of  saline  solu- 
tions, 135 

Solution,  135 

Solutions,  boiling  point  of,  155 

Sound,  velocity  of,  281 


336 


Index 


Specific  heat,  80-94 ;  effect  of 
change  of  state,  97 ;  effect  of 
temperature  and  pressure,  96  ;  of 
gases,  95  ;  table  of,  96  ;  true,  92' 

Specific  heats,  ratio  of,  281 

Spectrum,  248 

Spheroidal  state,  184 

Steam,  adiabatic  curve  for,  285  ; 
isothermal  for,  170;  latent  heat 
of,  1 60  ;  pressure  of,  172;  total 
heat  of,  159  ;  volume  of,  172 

Sublimation,  166 

Sulphur    dioxide,    liquefaction    of, 

175 
Superheated  steam,  173 

TEMPERATURE,  i  ;  absolute,  68  ; 
and  altitude,  316  ;  and  density, 
31  ;  and  level,  3  ;  of  the  air, 
316  ;  scales  of,  8 

Theory  of  exchanges,  261 

Theory  of  heat,  124 

Thermal  capacity,  83 

Thermal  units,  79 

Thermo-dynamics,  268 ;  first  law 
of,  123,  268  ;  second  law  of,  300 

Thermographs,  15 

Thermometers,  3  ;  air,  69  ;  alcohol, 
13;  differential,  1 6 ;  fixed  point 
of,  6 ;  maximum  and  minimum, 
13;  mercury,  5;  standard,  12; 
testing,  1 6  ;  weight,  35 

Thermometric  diffusivity,  216 

Thermo-pile,  226 

Thomson,  J.,  on  atmospheric  cir- 
culation, 322 ;  on  lowering  of 
freezing  point,  137,  138,  305 

Thomson,  Sir  William,  on  internal 
work  of  gases,  272 ;  scale  of  tem- 


peratures,   301  ;    second  law   of 
thermo-dynamics,  300 
Transmission  of  radiant  heat,  245 

UNITS,  C.G.S.,  no;  change  of, 
117,  260;  dynamical,  no;  of 
heat,  79 

VAPORIZATION,  144 ;  internal  and 
external  work  in,  162  ;  latent 
heat  of,  156  ;  in  vaciw^  145 

Vapour,  144 ;  at  maximum  pres- 
sure, 146 

Vapour-density,  185  ;  pressure,  146 

Viscous  solids,  141 

Volume,  change  of,  136 

WATER,  absolute  expansion  of,  47  ; 
conductivity,  221  ;  equivalent, 
84 ;  expansion  of,  50  ;  maximum 
density  of,  46  ;  specific  heat  of, 
94 

Water  vapour,  193 ;  condensation 
of,  31? 

Wave-motion,  230 

Weight  thermometer,  35 

Wet  and  dry  bulb  hygrometer,  200 

Wiedemann  and  Franz's  experi- 
ments, 211 

Williams's  (Major)  experiment,  137 

Wind,  320 ;  cause  of,  321,  322 ; 
trade,  322 

Work,  internal  and  external,  125, 
162  ;  of  gases,  275 

Work  done  by  a  gas  in  expanding, 
287 

ZERO,  absolute,  63 


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By  WILLIAM  JAGO,  F.C.S.,  F.I.C.  196  Experiments,  with  49  Wood- 
cuts, and  numerous  Questions  and  Exercises.  I2mo.  350  pages.  80 
cents. 

THE    PRINCIPLES    OF  CHEMISTRY. 

By  D.  MENDEL£EFF,  Professor  of  Chemistry  in  the  University  of  St.  Peters- 
burg. Translated  by  GEORGE  KAMENSKY,  A.R.S.M.,  of  the  Imperial 
Mint,  St.  Petersburg,  and  Edited  by  A.  J.  GREENAWAY,  F.I.C.,  Sub- 
Editor  of  the  Journal  of  the  Chemical  Society.  2  vols.  8vo.  1122 
pages.  $10.00. 

This  book  is  written  on  a  plan  essentially  different  from  that  of  the  ordinary 
text-books  on  chemistry.  The  text  of  the  work  deals  with  the  main  facts  and 
theories  of  the  science  in  such  a  way  as  to  be  readily  intelligible  to  a  student 
beginning  the  study  of  chemistry.  Supplementing  this,  and  intended  for  later 
study,  or  for  the  use  of  advanced  students,  are  the  numerous  footnotes,  in 
which  is  contained  the  bulk  of  the  work.  In  these  the  details  of  the  matter  re- 
ferred to  in  the  text  are  described,  and  on  debatable  points  the  views  held  by 
various  authorities  are  collated  and  discussed. 

"...  The  book  is  eminently  readable.  It  is  written  in  an  agreeable, 
almost  colloquial  style.  .  .  .  As  the  most  original  and  suggestive  treatise  on 
inorganic  chemistry  which  we  possess,  it  is  well  worthy  of  the  student's  atten- 
tion, and  must  be  regarded  as  a  very  important  addition  to  chemical  literature." 
—  The  Nation,  N.  Y. 

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SOLUTIONS. 

By  W.  OSTWALD,  Professor  of  Chemistry  in  the  University  of  Leipzig. 
Being  the  Fourth  Book,  with  some  additions,  of  the  second  edition  of 
Ostwald's  "Lehrbuch  der  Allgemeinen  Chemie."  Translated  by  M. 
M.  PATTISON  MUIR,  Fellow  of  Gonville  and  Caius  College,  Cambridge. 
8vo.  $3.00. 

"  The  work  of  translation  by  Mr.  Muir  is  excellent.  His  style  is  clear  and 
scholarly  throughout.  The  accuracy  of  the  translation  is  insured  by  the  fact 
that  Professor  Ostwald  has  revised  the  proofs  of  the  English  edition.  .  .  . 
This  translation  has  appeared  at  an  opportune  moment.  .  .  .  The  book 
can  be  heartily  recommended.  .  .  ."  — American  Chemical  Journal. 

"  The  work,  like  the  parent  treatise,  is  in  all  respects  admirable;  in  fact,  it 
would  be  difficult  to  point  out  a  more  suggestive  treatment  of  any  subject  in 
the  whole  domain  of  chemistry.  Those  chemists  who  do  not  read  German  will 
certainly  urge  the  translation  of  the  entire  work." — Nation,  N.  Y. 


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OUTLINES    OF   THEORETICAL   CHEMISTRY. 

By  LOTHAR  MEYER,  Professor  of  Chemistry  in  the  University  of  Tubingen. 
Translated  by  Professor  P.  PHILLIPS  BEDSON,  D.Sc.,  and  W.  CARLE- 
TON  WILLIAMS,  B.  Sc.  8vo.  232  pages.  $2. 50  net. 

This  book  gives  a  concise  account  of  the  theories  of  modern  chemistry, 
which  it  is  expected  will  not  only  be  of  use  to  advanced  students,  but  will 
enable  those  who  take  a  general  interest  in  science,  but  are  unfamiliar  with  the 
details  of  chemical  investigation,  to  gain  a  general  idea  of  the  development  of 
theoretical  chemistry. 

INTRODUCTION  TO  THE  STUDY  OF  CHEMICAL  PHI- 
LOSOPHY. The  Principles  of  Theoretical  and  Systematic  Chemis- 
try. 

By  WILLIAM  A.  TILDEN,  B.Sc.  (London),  F.C.S..  Professor  of  Chemistry 
in  the  Mason  College,  Birmingham.  With  Woodcuts.  (TEXT-BOOKS 
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and  Answers  specially  adapted  for  Use  in  Colleges  and  Science  Schools. 

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turer in  Chemistry  in  the  University  College,  Nottingham.  With  a 
Preface  by  Professor  F.  CLOWES,  D.Sc.  (Lond.),  F.I.C.  I2mo.  60 
cents. 

This  little  book  has  been  written  to  supply  the  need  of  a  work  on  Chemical 
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and  yet  explicit  account  of  the  methods  of  solving  them.  It  is  not  written  for 
the  use  of  candidates  preparing  for  any  special  examinations.  It  is  intended  to 
form  a  part  of  the  course  of  teaching  or  study  suitable  to  the  chemical  student 
who  wishes  to  equip  himself  for  the  various  duties  which  any  well-trained 
chemist  may  be  called  upon  to  perform. — Author 's  Preface. 

*x*  ".     .     .     A  well  planned  text-book." — Science,  N.   Y. 

THE  METHODS  OF  GLASS-BLOWING.  For  the  Use  of  Physi- 
cal and  Chemical  Students. 

By  W.  A.  SHENSTONE,  Lecturer  on  Chemistry  in  Clifton  College.  With  42 
Illustrations.  Crown  8vo.  96  pages.  50  cents. 

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try," THORPE'S  "  Dictionary  of  Applied  Chemistry,"  etc.,  see  Longmans,  Green, 
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ELECTRICAL     ENGINEERING     FOR      ELECTRIC     LIGHT 

ARTISANS  AND  STUDENTS. 
By  W.  SLINGO,  Principal  of  the  Telegraphists'  School  of  Science,  and  A. 

BROOKER,  Instructor  on  Electrical  Engineering  at  the  Telegraphists' 

School  of   Science.     With  307  Illustrations.     Crown  8vo.     640  pages. 

$3-50. 

Contents. — I.  Current  —  Potential  —  Conductors — Insulators.  II.  Practical 
Units — Ohm's  Law — Fundamental  Units.  III.  Primary  Batteries.  IV.  Meas- 
urement of  Current  Strength.  V.  Measurement  of  Resistance.  VI.  Measure- 
ment of  Electro  -  Motive  Force.  VII.  Electro  -  Magnets  —  Electro  -  Magnetic 
Induction.  VIII.  Dynamo-Electric  Machines  (Alternate  Current).  IX.  Dyna- 
mo Electric  Machines  (Direct  Current).  X.  Direct  Current  Dynamos — Con- 
tinued. XI.  Direct  Current  Dynamos  (Open  Coil).  XII.  Motors  and  their  Ap- 
plications. XIII.  Transformers.  XIV.  Secondary  Batteries.  XV.  Arc  Lamps. 
XVI.  Incandescent  Lamps — Photometry.  XVII.  Installation  Equipment,  Fit- 
tings, etc.  Index. 

"It  is  as  complete  as  anything  we  have  ever  seen.  It  should  meet  with  a 
hearty  reception  among  electricians  and  students  of  electricity,  for  it  is  one  of 
the  most  comprehensive  works  ever  published.  Everything  that  is  necessary 
to  a  clear  understanding  of  electric  lighting  and  kindred  subjects  is  found  in 
this  volume,  and  we  think  that  every  individual  of  the  classes  mentioned  would 
greatly  further  his  own  interests  by  possessing  and  studying  this  work." — The 
Electric  Age. 

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lishers. 

TELEPHONE  LINES  AND  THEIR  PROPERTIES. 

By  Prof.  W.  J.  HOPKINS,  of  the  Drexel  Institute,  Philadelphia.  With 
numerous  Illustrations.  I2mo.  [March,  1893. 

The  intention  of  the  author  has  been  to  provide  a  book  which  should  prove 
useful  to  the  practical  man,  as  well  as  one  which  would  serve  as  a  basis  for  a 
lecture  course  to  students.  He  has  therefore  thought  it  desirable  to  devote 
some  space  to  the  development  of  correct  elementary  ideas  of  matter  and 
energy  so  as  to  lead  up  to  the  most  modern  conception  of  the  method  of  prop- 
agation of  electro-magnetic  disturbances. 

ELECTRICITY  AND  MAGNETISM. 

By  FLEEMING  JENKIN,  F.R.SS.L.  and  E.,  M.I.C.E.,  late  Professor  of  En- 
gineering in  the  University  of  Edinburgh.  With  177  Illustrations. 
(TEXT-BOOKS  OF  SCIENCE.)  i2mo.  415  pages.  $1.25. 

Contents.— Electric  Quantity,  Potential,  Current,  Resistance,  Electro-Static 
Measurement,  Magnetism,  Magnetic  Measurements,  Electro-Magnetic  Meas- 
urement, Units,  Chemical  Theory  of  Electro-Motive  Force,  Thermo-Electric- 
ity,  Galvanometers,  Electrometers,  Galvanic  Batteries,  Frictional  Electrical 
Machines,  Electro-Magnetic  Engines,  Telegraphic  Apparatus,  Speed  of  Sig- 
nalling. Telegraphic  Lines,  Useful  Applications  of  Electricity,  Mariners'  Com- 
pass, etc.,  etc.  Appendix  on  the  Telephone  and  Microphone  and  27  Tables. 


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MAGNETISM  AND  ELECTRICITY.      A  Manual  for  Students  in 

Advanced  Classes. 
By  A.  W.   POYSER,  M.A. ,  Trinity  College,  Dublin.     With  over  300  figures 

in  the  text.      I2mo.     394  pages.      $1.50. 

This  book  is  written  on  the  same  plan  as  the  author's  Elementary  work,  and 
it  is  hoped  that  it  may  prove  of  assistance  to  students  who  are  desirous  of  ob- 
taining an  experimental  knowledge  of  the  facts  and  laws  of  the  science  of  Mag- 
netism and  Electricity. 

The  book  has  been  thrown  into  experimental  form  for  several  reasons,  of 
which  two  may  be  mentioned  :  (i)  experimental  work,  apart  from  the  actual 
knowledge  gained,  affords  a  valuable  training  to  the  mind,  inasmuch  as  a  stu- 
dent acquires  the  habit  of  making  careful  observations,  and  drawing  inferences 
from  facts  ;  and  (2)  scientific  knowledge  obtained  merely  from  book-work,  with 
a  view  of  passing  a  particular  examination,  is  almost  worse  than  useless,  and 
may  indeed  defeat  the  object  at  which  the  student  is  aiming. 

A  series  of  exercises,  containing  many  numerical  problems,  has  been  in- 
terspersed throughout  the  text,  and  forms  an  important  feature  of  the  book. 
The  student,  who  is  assumed  by  the  author  to  have  read  the  elements  of  Al- 
gebra, Geometry,  and  Trigonometry,  is  recommended  by  him  to  systematically 
work  these  exercises,  as,  in  his  opinion,  the  application  of  Mathematics  is  ab- 
solutely essential  in  order  to  obtain  a  thorough  grasp  of  any  subject  in  Physical 
Science. 

A  short  account  of  some  of  the  practical  applications  of  Electricity  has  been 
given  ;  and  in  order  that  the  reader  may  form  some  idea  of  the  direction  of 
modern  thought  in  the  science,  a  chapter  has  been  added,  at  the  end  of  the 
book,  on  recent  researches,  which  may  prove  an  incentive  to  further  study. 

MAGNETISM  AND  ELECTRICITY. 

By  A.  W.  POYSER,  B.A.,  Trinity  College,  Dublin.  With  235  Diagrams  and 
Illustrations.  I2mo.  255  pages.  80  cents. 

This  volume  is  intended  as  an  introduction  for  beginners  in  these  subjects, 
and  covers  the  course  usually  taken  in  a  year's  school  work.  The  book  is  the 
result  of  practical  experience  in  teaching,  and  it  has  been  thrown  into  experi- 
mental form  from  a  conviction  that,  if  the  student  is  to  gain  an  adequate 
knowledge  of  the  subject,  it  is  absolutely  necessary  for  him  to  acquire  it  by  ex- 
periment. Exercises  are  interspersed  throughout  the  book  in  addition  to  the 
miscellaneous  examples  on  page  238.  Numerical  examples  are  frequently 
given,  as,  even  in  the  most  elementary  work,  the  student  should  learn  that  some 
knowledge  of  mathematics  is  not  only  useful  but  essential. 

"  Beyond  question  one  of  the  best  text-books  on  this  important  subject  yet 
presented  to  the  American  public." — Education,  Boston. 

ELECTRICITY  FOR  SCHOOLS  AND  COLLEGES. 

By  W.  LARDEN,  M.A.,  author  of  "A  School  Course  in  Heat,"  in  Use  at 
Rugby,  Clifton,  Cheltenham,  Bedford,  Birmingham,  King's  College, 
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and  a  Series  of  Examination  Papers  with  Answers.  I2mo.  528  pages. 
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ELECTRICITY  TREATED  EXPERIMENTALLY.     For  the  Use 

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EXERCISES  IN  ELECTRICAL  AND  MAGNETIC  MEAS- 
UREMENTS, 

By  R.  E.  DAY,  M.A.,  King's  College,  London.     i2mo.    204  pages.    $1.00. 

The  object  of  this  book  is  to  lay  before  the  student,  under  the  form  of  Prob- 
lems, Numerical  Illustrations  of  the  Main  Facts  of  Electricity  and  Magnetism, 
with  especial  reference  to  the  Modern  Doctrine  of  Energetics. 

POTENTIAL,  AND    ITS    APPLICATION   TO  THE    EXPLA- 
NATION OF  ELECTRICAL   PHENOMENA.     Popularly  treated. 

By  Dr.  TUMLIRZ,  Lecturer  in  the  German  University  of  Prague.  Translated 
by  D.  ROBERTSON,  M.A.,  LL.B.,  B.Sc.,  formerly  Assistant  Master  at 
University  College  School.  With  108  Illustrations.  Crown  8vo.  284 
pages.  $1.25. 

THE  ART  OF  ELECTRO-METALLURGY,  including  all  Known 
Processes  of  Electro-Deposition. 

By  G.  GORE,  LL.D.,  F.R.S.  With  56  Woodcuts.  (TEXT- BOOKS  OF  SCI- 
ENCE.) I2mo.  418  pages.  $2.00. 

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OPTICAL  PROJECTION  :  a  Treatise  on  the  Use  of  the  Lantern  in  Ex- 
hibition and  Scientific  Demonstration. 

By  LEWIS  WRIGHT,  author  of  "  Light  ;  a  Course  of  Experimental  Optics." 
With  232  Illustrations.  I2mo.  438  pages.  $2.25. 

"Mr.  Wright's  book  gives  all  that  is,  at  present,  at  least,  necessary  for  a 
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lantern  rests.  .  .  .  The  book  is  very  full  of  useful  detail,  and  is  eminently 
practicable.  .  .  .  Will  assuredly  be  warmly  welcomed  by  teachers  and 
lecturers."— Nation,  N.  Y. 

"  A  practical  guide  for  those  who  wish  to  gain  exact  knowledge  about  pro- 
jections, and  instruments  for  such  work.  We  commend  its  perusal  to  those 
interested  in  any  line  of  work  requiring  lantern  illustrations." — Sidereal  Mes- 
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ELEMENTARY    PHYSICS. 

By  MARK  R.  WRIGHT,  author  of  "Sound,  Light,  and  Heat."  With  238 
Illustrations.  J2mo.  256  pages.  80  cents. 

This  work  will  serve  as  a  suitable  text-book  for  any  class  beginning  the 
study  of  physics.  The  leading  facts  are  brought  under  the  notice  of  the  stu- 
dent by  easy  experiments  that  do  not  demand  expensive  apparatus.  Full  in- 
structions are  given  for  the  construction  of  the  apparatus,  in  the  text,  or  in  the 
appendix.  The  author  believes  that  in  early  lessons  it  is  inadvisable  to  trouble 
the  student  either  with  theories  or  with  the  generalizations  that  prove  such  a 
valuable  aid  to  the  advanced  student. 

BOWDOIN  COLLEGE. 

"  I  consider  it  very  well  adapted  to  beginners  in  the  science.  The  long  list 
of  experiments  affords  ample  material  from  which  a  course  of  any  length  may 
be  adapted.  The  whole  trend  of  the  book  is  in  the  right  direction — the  teaching 
of  physics  by  experiment." — C.  C.  HUTCHINS,  Brunswick,  Me. 

"  It  should  take  high  rank,  and  come  into  wide  use  among  our  schools." — 
Education,  Boston. 

"This  book  possesses  considerable  merit.  The  matter  contained  in  it  is 
just  about  as  much  as  would  cover  the  course  usually  taken  in  a  year's  school 
work  ;  the  explanatory  text  is  couched  in  the  clearest  language,  and  the  ex- 
periments described  are  capable  of  being  easily  brought  to  a  successful  termi- 
nation. .  .  .  The  book  compares  most  favorably  with  any  written  for  the 
purpose  of  imparting  a  rudimentary  knowledge  of  magnetic  and  electrical  phe- 
nomena. ' ' — Nature. 

"The  treatment  is  simple  and  clear,  and  the  experiments  given  are  admir- 
ably chosen  ;  by  their  aid  the  subject  is  logically  developed,  and  they  are  such 
as  can  be  performed  with  very  simple  apparatus.  .  .  .  Altogether,  this  is 
one  of  the  best  primary  books  on  physics  that  we  have  seen." — Critic,  New 
York. 

PRACTICAL  PHYSICS. 

By  R.  T.  GLAZEBROOK,  M.A.,  F.R.S.,  and  W.  N.  SHAW,  M.A.,  Demon- 
strators at  the  Cavendish  Laboratory,  Cambridge.  With  Woodcuts. 
(TEXT-BOOKS  OF  SCIENCE.) 

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cal Laboratories.  It  contains  chapters  on  Physical  Measurements,  Units  of 
Measurements,  Physical  Arithmetic,  Measurement  of  the  more  Simple  Quanti- 
ties, Measurement  of  Mass  and  Determination  of  Specific  Gravities,  Mechanics 
of  Solids,  Liquids,  and  Gases,  Acoustics,  Thermometry,  and  Expansion.  Cal- 
orimetry,  Tension  of  Vapor  and  Hygrometry,  Photometry,  Mirrors  and  Lenses, 
Spectra,  Refractive  Indices  and  Wave-Lengths,  Polarized  Light,  Color  Vision, 
Magnetism,  Electricity,  Experiments  in  the  Fundamental  Properties  of  Electric 
Currents,  Ohm's  Law,  Galvanometric  Measurement  of  a  Quantity  of  Electric- 
ity, etc.,  etc. 

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SOUND,    LIGHT,    AND    HEAT. 

By  MARK  R.  WRIGHT,  author  of  ' '  Elementary  Physics."     With  162  Illus- 
trations and  Diagrams.     I2mo.     272  pages.     80  cents. 

This  volume  is  an  elementary  text-book  on  Sound,  Light,  and  Heat,  treated 
experimentally.  It  is  essential  that  the  experiments  should  be  performed. 
The  numerical  results  which  illustrate  the  text  should  not  take  the  place  of 
measurements  made  by  the  student.  The  experiments  demand  no  expensive 
apparatus ;  the  aim  has  been  to  avoid  elaborate  instruments  ;  descriptions  of 
the  apparatus  used  appear  in  the  book  or  in  the  appendix.  Examples  are 
given  frequently  throughout  the  work. 

HEAT. 

A  Manual  for  Students  in  Advanced  Classes. 

By  MARK  R.  WRIGHT.     With  numerous  Illustrations  and  Diagrams.  Crown 
8vo.     [Nearly  Ready.] 


PHYSICAL   OPTICS. 

By  R.  T.  GLAZEBROOK,  M.A.,  F.R.S.,  Fellow  and  Lecturer  of  Trinity 
College ;  Demonstrator  of  Physics  at  the  Cavendish  Laboratory,  Cam- 
bridge. With  183  Woodcuts  of  Apparatus,  etc.  (TEXT-BOOKS  OF 
SCIENCE.)  I2mo.  448  pages.  $2.00. 

Contents. — Wave  Motion,  The  Rectilinear  Propagation  of  Waves,  Reflection 
and  Refraction,  Prisms  and  Lenses,  Interference,  Colors  of  Thin  Plates,  Dif- 
fraction, Dispersion  and  Achromatism,  Spectrum  Analysis,  Absorption  and 
Anomalous  Dispersion,  Double  Refraction,  Refraction  and  Reflection  of  Polar- 
ized Light,  Interference  of  Polarized  Light,  Circular  Polarization,  Electro- 
Optics,  The  Velocity  of  Light,  etc.,  etc. 

THEORY    OF  HEAT. 

By  J.  CLERK  MAXWELL,  M.A.,  etc.,  etc.  Tenth  Edition,  with  Corrections 
and  Additions  by  LORD  RAYLEIGH,  Sec.  R.  S.  (TEXT-BOOKS  OF 
SCIENCE.)  I2mo.  357  pages.  $1.50. 

The  aim  of  this  book  is  to  exhibit  the  scientific  connection  of  the  various 
steps  by  which  our  knowledge  of  the  Phenomena  of  Heat  has  been  extended. 
It  treats  of  Thermometry,  Calorimetry,  Lines  of  Equal  Temperature  on  the 
Indicator  Diagram,  Adiabatic  Lines,  Heat  Engines,  Latent  Heat,  Thermo- 
dynamics of  Gases,  Free  Expansion,  Determination  of  Heights  by  the  Barom- 
eter, Radiation,  Convection  Currents,  Diffusion  of  Heat  by  Conduction, 
Diffusion  of  Fluids,  Capillarity,  Elasticity  and  Viscosity,  Molecular  Theory  of 
the  Constitution  of  Bodies,  etc.,  etc. 

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Green,  &*  Co. 's  Educational  Catalogue. 


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LONGMANS,   GREEN,   &  CO.' 'S  PUBLICATIONS. 

LESSONS  IN  ELEMENTARY  SCIENCE. 
LONGMANS'    OBJECT    LESSONS.     Hints  on    Pre- 
paring and  Giving  Them.     With  full  Notes  of  Com- 
plete Courses  of  Lessons  on  Elementary  Science. 
By  DAVID  SALMON,  Principal  of  the  Training  College,  Swansea  ;  Revised 
and  Adapted  to  American  Schools  by  JOHN  F.  WOODHULL,  Professor 
of  Methods  of  Teaching  Natural  Science  in  the  New  York  College  for 
the   Training   of   Teachers.      I2mo.      246  pages.       152    Illustrations. 
Mailing  Price,  $1.10. 

PART   I. — HINTS   ON    PREPARING   AND    GIVING    LESSONS. 

Should  Science  be  Taught? — When  should  Science  Teaching  Begin ? — 
Subjects  of  Lessons — Matter  of  Lessons — Notes  of  Lessons — Illustrations — 
Language — Questions — Telling  and  Eliciting — Emphasis — Summary — Re- 
capitulation. (Pp.  1-36.) 

PART    II. — NOTES    OF   LESSONS. 

First  Year. — (a)  Lessons  on  Common  Properties,  (b)  Lessons  on  Common 
Animals,  (c)  Lessons  on  Plants. 

Second  Year. — (a)  Lessons  on  Common  Properties,  (b)  Lessons  on  Animals. 
(c)  Lessons  on  Plants. 

Third  Year. — (a)  Lessons  on  Elementary  Chemistry  and  Physics,  (b)  Les- 
sons on  Animals,  (c)  Lessons  on  Flowers. 

Fourth  Year. — (a)  Lessons  on  Elementary  Physics,  (b)  General  Lessons  on 
Natural  History,  (c)  Lessons  on  Elementary  Botany. 

Notes  of  a  Lesson  on  the  Cat. — Index.     (Pp.  41-238.) 

"  If  these  lessons  are  given  at  the  rate  of  one  a  week,  and  thoroughly  re- 
viewed from  time  to  time,  they  will  provide  work  for  four  or  five  years.  Teach- 
ers and  pupils  should  make  large  use  of  cyclopedias  and  other  sources  of  in- 
formation. Hence  the  book  offers  a  course  of  elementary  science  for  lower 
grades,  leading  up  to  the  specific  study  of  zoology,  physiology,  botany,  chem- 
istry, physics,  and  geology,  which  are  to  be  undertaken  in  the  higher  grades." 

' '  A  four  years'  course  in  science  is  here  scheduled  that  embraces  botany, 
zoology,  chemistry,  and  physics.  The  four  subjects  are  studied  throughout  the 
course,  the  lessons  being  graded  to  suit  the  stage  of  intellectual  development  of 
the  child.  The  plan  adopted  is  eminently  objective  and  inductive.  .  .  . 
throughout  the  book  new  knowledge  gained  is  made  the  stepping-stone  to  some- 
thing higher,  co-ordinating  not  only  the  facts  of  any  one  science  but  also  the 
various  sciences  themselves.  The  process  of  comparing  objects,  in  order  to  de- 
termine their  similarities  and  differences  as  a  basis  of  classification,  is  most  ad- 
mirably developed.  .  .  .  Manuals  heretofore  have,  as  a  general  rule,  treated 
each  object  as  if  it  were  isolated  from  all  else  in  the  material  world,  and  as  if 
the  facts  concerning  that  particular  object  were  of  prime  importance.  This 
book  subordinates  the  knowledge  gained  of  particular  objects  to  the  use  of  ob- 
jects as  a  means  of  exercising  the  powers  of  observation,  comparison,  and  gen- 
eralization."— Educational  Review,  N.  Y. 


LONGMANS,  GREEN,  &  CO.,  15  East  Sixteenth  Street,  New  York. 


LONGMANS,   GREEN,   or=  CO.' S  PUBLICATIONS. 

THE    ELEMENTS    OF   MECHANISM. 

By  T.  M.  GOODEVE,  M.A.  With  342  Woodcuts.  New  Edition,  Revised 
and  Enlarged.  Crown  8vo.  350  pages.  $2.00. 

Contents. — Introductory  :  On  the  Conversion  of  Circular  into  Reciprocating 
Motion,  On  Linkwork,  On  the  Conversion  of  Reciprocating  into  Circular  Mo- 
tion, On  the  Teeth  of  Wheels,  On  the  Use  of  Wheels  in  Trains,  Aggregate 
Motion.  On  the  Truth  of  Surface,  and  the  Power  of  Measurement,  Miscel- 
laneous Contrivances,  etc. 

PRINCIPLES    OF    MECHANICS. 

By  T.  M.  GOODEVE,  M.A.  New  Edition,  Re- written  and  Enlarged.  With 
253  Woodcuts.  Crown  8vo.  358  pages.  $2.00. 

In  this  volume  an  endeavor  has  been  made  to  present  a  comprehensive  view 
ol  the  Science  of  Mechanics,  to  point  out  the  necessity  of  continually  referring 
to  practice  and  experience,  and  above  all  to  show  that  the  relation  of  the  theory 
of  heat  to  mechanics  should  be  approached  by  the  student,  in  his  earliest  in- 
quiries, with  the  same  careful  thought  with  which  he  will  regard  it  when  his 
knowledge  has  become  more  extended. 

LESSONS  IN  ELEMENTARY  MECHANICS.  Introductory  to 
the  Study  of  Physical  Science.  With  Numerous  Exercises. 

By  Sir  PHILIP  MAGNUS.  New  Edition,  Re-written  and  Enlarged.  (Thir- 
tieth thousand.)  I2mo.  387  pages.  Mailing  price,  $1.20. 

"  The  special  feature  of  this  book  is  the  admirable  manner  in  which  energy 
is  discussed  and  its  operation  illustrated,  ...  an  excellent  high  school 
course. " — Science. 

THEORETICAL  MECHANICS,  INCLUDING  HYDROSTAT- 
ICS AND  PNEUMATICS. 

ByJ.  E.  TAYLOR,  M.A.,  Hon.  Inter.  B.Sc.,  Central  High  Schools,  Shef- 
field. With  175  Diagrams  and  Illustrations,  and  522  Examination 
Questions  and  Answers.  I2mo.  272  pages.  80  cents. 

"  As  a  combination  of  the  best  methods,  best  illustrations,  and  best  exam- 
ples, it  is  a  great  success." — Journal  of  Education. 

ELEMENTS  OF  MACHINE  DESIGN. 

An  Introduction  to  the  Principles  which  determine  the  Arrangement  and 
Proportion  of  the  Parts  of  Machines,  and  a  Collection  of  Rules  for  Ma- 
chine Designs. 

By  W.  CAWTHORNE  UNWIN,  B.Sc.,  Assoc.  Inst.  C.E.  Eleventh  Edition, 
Revised  and  Enlarged. 

Part  I.  General  Principles,  Fastenings,  and  Transmissive  Machinery. 
With  304  Diagrams  and  Illustrations.  Crown  8vo.  476  pages.  $2.00. 

Part  II.     Chiefly  Engine  Details.     305  pages.     $1.50. 


LONGMANS,  GREEN,  &  CO.,  15  East  Sixteenth  Street,  New  York. 


LONGMANS,   GREEN,   &  CO.  'S  PUBLICATIONS. 

LESSONS   IN    ELEMENTARY    MECHANICS. 

By  W.  H.  GRIEVE,  P.S.A.,  late  Engineer  R.N.,  Science  Demonstrator  for 
the  London  School  Board,  etc. 

Part  i.  Matter  in  three  states:  solids,  liquids,  gases.  Mechanical  proper- 
ties peculiar  to  each  state.  Matter  is  porous,  compressible,  elastic. 
Measurement  as  practised  by  mechanics.  Measures  of  length,  time, 
velocity,  and  space.  With  165  Illustrations  and  a  large  number  of  Ex- 
amples. I2mo.  50  cents. 

Part  2.  Matter  in  motion.  The  weight  of  a  body,  its  inertia  and  momen- 
tum. Measures  of  force,  work,  and  energy.  Energy  may  be  trans- 
formed, but  cannot  be  destroyed.  Heat  as  a  form  of  energy.  With  122 
Illustrations.  I2mo.  50  cents. 

Part  3.  The  simple  mechanical  powers,  viz.:  (i)  the  lever  ;  (2)  the  wheel 
and  axle  ;  (3)  pulleys  ;  (4)  the  inclined  plane  ;  (5)  the  wedge  ;  (6)  the 
screw.  Liquid  pressure  ;  the  hydrostatic  press  ;  liquids  under  the  action 
of  gravity.  The  parallelogram  of  forces  ;  the  parallelogram  of  veloci- 
ties ;  examples  commonly  met  with  illustrating  the  mechanical  powers. 
With  103  Illustrations.  I2mo.  50  cents. 

ELEMENTARY  SCIENCE  LESSONS.     Being  a  Systematic  Course 

of  Practical  Object  Lessons.      Illustrated  by  Simple  Experiments. 
By  W.  HEWITT,  B.Sc.     Parts  L,  II.,  III.,  and  IV.     Each,  50  cents. 

AN  INTRODUCTION  TO  MACHINE-DRAWING  AND  DE- 
SIGN. 

By  DAVID  ALLAN  Low  (Whitworth  Scholar),  Principal  of  the  Technical 
School,  People's  Palace,  London.  With  65  Illustrations  and  Dia- 
grams. Crown  8vo.  60  cents. 

In  producing  this  work  the  author  has  aimed  at  placing  before  young  stu- 
dents and  others,  who  wish  to  acquire  the  skill  and  knowledge  necessary  for 
making  the  simpler  working  drawings,  such  as  are  produced  in  engineers'  draw- 
ing offices,  a  number  of  good  exercises  in  drawing,  sufficient  for  one  session's 
work,  and  at  the  same  time  a  corresponding  amount  of  information  on  the  de- 
sign of  machine  details  generally. 

EXERCISES  IN  WOOD  WORKING  for  Handicraft  Classes  in  Ele- 
mentary and  Technical  Schools. 

By  WILLIAM  CAWTHORNE  UNWIN,  F.R.S.,  Member  Inst.  C.E.  28  Plates. 
Fcp.  folio.  In  case.  $1.50. 

V*  For  other  books  on  elementary  science,  see  Longmans,  Green,  &  Co.'s  Cata- 
logue of  Educational  Works. 


LONGMANS,  GREEN,  &  CO.,  15  East  Sixteenth  Street,  New  York. 


LONGMANS,   GREEN,   &  CO: S  PUBLICATIONS. 

BUILDING  CONSTRUCTION.     A  Manual  for  Students. 
By  the  Author  of  "Notes  on  Building  Construction."     Crown  8vo.     "With 
nearly  400  Illustrations  and  Index.     $1.50. 

"  To  students  who  have  advanced  beyond  the  elementary  stage  it  supplies  a 
wide  range  of  information  in  the  form  of  succinct  notes,  with  illustrations  oV 

nearly  everything  that  can  be  illustrated  to  any  real  purpose." Architectural 

Record, 

BUILDING  CONSTRUCTION. 

By  EDWARD  J.  BURRELL,  Teacher  of  Building  Construction  at  the  Tech- 
nical School  of  the  People's  Palace,  Mile  End.  Fully  Illustrated,  with 
303  Working  Drawings.  I2mo.  256  pages.  80  cents. 

The  chief  aim  of  the  writer  of  this  book  has  been  to  place  before  the  student 
numerous  examples  of  constructive  details,  which  shall  not  only  serve  as  illustra- 
tions to  the  text,  but  shall  also  afford  the  data  necessary  for  making  scale  draw- 
ings of  the  various  parts.  With  this  end  in  view  the  diagrams  have  been  care- 
fully dimensioned. 

ON  THE  STRENGTH  OF  MATERIALS  AND  STRUCT- 
URES  ;  the  Strength  of  Materials,  as  depending  on  their  quality  and 
as  ascertained  by  Testing  Apparatus  ;  the  Strength  of  Structures,  as 
depending  on  their  form  and  arrangement,  and  on  the  materials  of 
which  they  are  composed. 

By  Sir  JOHN  ANDERSON,  C.E.,  LL.D.,  F.R.S.E.  Ninth  Edition,  with 
66  figures.  (TEXT- BOOKS  OF  SCIENCE.)  i2mo.  322  pages.  $1.25. 

THE  THEORY  OF  STRESSES  IN  GIRDERS  AND  SIMILAR 
STRUCTURES.  With  Practical  Observations  on  the  Strength  and 
other  Properties  of  Materials. 

By  BINDON  B.  STONEY,  LL.D.,  F.R.S.,  M.I.C.E.  With  5  Plates  and  143 
Illustrations  in  the  Text.  Royal  8vo.  786  pages.  $12.50. 

THERMODYNAMICS. 

By  RICHARD  WORMELL,  D.Sc.,  M.A.  Second  Edition.  I2mo.  180  pages 
50  cents. 

"  The  present  text-book  is  an  attempt  to  do  for  Thermodynamics  what  has 
long  since  been  done  for  Elementary  Dynamics.  The  work  aims  at  tracing,  in 
a  systematic  manner,  the  reasoning  by  which  the  Dynamical  Theory  of  Heat, 
and  its  chief  consequences,  are  established.  Each  chapter  opens  with  a  brief 
description  of  some  experiments  required  either  to  supply  the  fund  of  observa- 
tions on  which  the  laws  and  theory  are  founded,  or  to  show  how  certain  con- 
stant quantities  used  in  the  calculations  have  been  determined  by  careful  and 
trustworthy  physicists.  Then  follows,  first,  a  discussion  or  examination  of  the 
experiments  ;  secondly,  the  definitions  of  the  scientific  terms  required  for  the 
expression  of  the  general  laws  to  which  they  conduct  us  ;  next,  we  have  the 
formal  enunciation  of  these  laws  ;  and  finally  the  strictly  logical  or  mathemati- 
cal consequences  which  can  be  deducted  from  them.'' — From  Preface. 


LONGMANS,  GREEN,  &  CO.,  15  East  Sixteenth  Street,  New  York. 


LONGMANS,   GREEN,   6°  CO.' 'S  PUBLICATIONS. 

STEAM. 

By  W.  RIPPER,  M.I.M.E.  Principal  of  the  Technical  Schools,  Sheffield. 
With  142  Illustrations.  I2mo.  210  pages.  80  cents. 

"  A  work  of  great  utility  to  students  of  engineering,  and  especially  of  steam 
power  and  its  application." — Engineer. 

11  Many  civil  engineers  whose  practice  is  not  directly  connected  with  steam 
engineering  feel  the  need  of  just  such  a  book,  and  we  know  of  none  which  we 
could  recommend  to  them  with  more  confidence  for  such  a  purpose." 

— Engineering  News. 

THE    STEAM   ENGINE. 

By  GEORGE  C.  V.  HOLMES,  Whitworth  Scholar ;  Secretary  of  the  Institu- 
tion of  Naval  Architects.  With  212  Woodcuts.  (TEXT-BOOKS  OF 
SCIENCE.)  i2mo.  544  pages.  $2.00. 

HYDROSTATICS    AND    PNEUMATICS. 

By  Sir  PHILIP  MAGNUS.     79  Diagrams.     I2mo.      182  pages.     50  cents. 
With  Answers.     65  cents. 

GRAPHICS;  OR,  THE  ART  OF  CALCULATION  BY 
DRAWING  LINES,  applied  especially  to  Mechanical  Engineering. 

By  ROBERT  H.  SMITH,  Professor  of  Engineering,  Mason  College,  Birming- 
ham, etc.  Part  I.  Arithmetic,  Algebra,  Trigonometry,  Moments, 
Vector  Addition,  Locor  Addition,  Machine  Kinematics,  and  Statics  of 
Flat  and  Solid  Structures.  With  Separate  Atlas  of  29  Plates,  contain- 
ing 97  Diagrams.  8vo.  282  pages.  Price  (Text  and  Atlas),  $5.00. 

THE  STEPPING-STONE  TO  ARCHITECTURE:  Explaining 
in  simple  language  the  Principles  and  Progress  of  Architecture  from  the 
earliest  times. 

By  THOMAS  MITCHELL.  Illustrated  by  engravings  (49  figures  and  22 
plates).  i8mo.  50  cents. 

A  GRADUATED  COURSE  OF  SIMPLE  MANUAL  TRAIN- 
ING EXERCISES  FOR  EDUCATING  THE  HAND  AND 
EYE. 

By  W.  HEWITT,  B.Sc.,  Science  Demonstrator  for  the  Liverpool  School 
Board.  Part  I.  First  and  Second  Series.  With  four  colored  Plates 
and  numerous  Diagrams.  Crown  8vo.  80  cents. 


LONGMANS,  GREEN,  &  CO.,  15  East  Sixteenth  Street,  New  York. 


LONGMANS,   GREEN,   &  CO.1 S  PUBLICATIONS. 

GEOMETRICAL  DRAWING  FOR  ART  STU- 
DENTS. Embracing  Plane  Geometry  and  its  Ap- 
plication, the  Use  of  Scales,  and  the  Plans  and  Eleva- 
tions of  Solids.  With  nearly  600  Figures. 

By  I.  HAMMOND  MORRIS.     Crown  8vo.     192  pages.     60  cents. 

This  book  contains  the  first  sixteen  chapters  of  the  author's  "  Practical  Plane 
and  Solid  Geometry,"  with  Examination  paper  added. 

*^For  other  books  on  Geometry,  Trigonometry,  Calculus,  etc. ,  see  Longmans, 
Green,  &*  CoSs  Catalogue  of  Educational  Works. 

A    HANDBOOK   OF    CRYPTOGAMIC    BOTANY. 

By  A.  W.  BENNETT,  M.A.,  B.Sc.,  F.L.S.,  Lecturer  on  Botany  at  St. 
Thomas's  Hospital,  and  GEORGE  R.  MILNE  MURRAY,  F.L.S.,  Nat- 
ural History  Department,  British  Museum.  With  378  Illustrations. 
8vo.  481  pages.  $5.00. 

No  general  handbook  of  Cryptogamic  Botany  has  appeared  in  the  English 
language  since  Berkeley's,  published  in  1857.  The  present  volume  gives  de- 
scriptions of  all  the  classes  and  more  important  orders  of  Cryptogams,  includ- 
ing all  the  most  recent  discoveries  and  observations. 

ELEMENTARY  TEXT-BOOK  OF  BOTANY.    For 

the  use  of  Schools. 

By  EDITH  AITKEN,  late  Scholar  of  Girton  College,  and  Certificated  Student 
in  First  Class  Honors  of  the  University  of  Cambridge.  With  131  Illus- 
trations and  Index.  I2mo.  262  pages.  $1.50. 

"  A  good  teacher  with  a  small  class  of  bright  and  willing  students  would  find 
not  only  profit  but  much  pleasure  in  pursuing  Miss  Aitken's  System.  .  ." 
—  The  Nation,  N.  Y. 

"  Clear,  methodical,  and  thoroughly  practical." — Epoch. 

ELEMENTARY  BOTANY,  THEORETICAL  AND 
PRACTICAL. 

By  HENRY  EDMONDS,  B.Sc.  London.  New  and  Revised  Edition.  With 
319  Diagrams  and  Woodcuts.  I2mo.  220  pages.  80  cents. 

This  book  contains  chapters  on  Structure  of  the  Seed,  Cell  Structure,  Cell 
Growth,  Shape  and  Formation,  Germination,  Root.  Growth,  Structure  and 
Functions,  Stem  Structure  and  Functions,  Buds  and  Ramification,  Leaf  Struct- 
ure and  Functions,  Bracts  and  Inflorescence,  Flower  Structure  and  Functions, 
Fruit  and  Seed,  Movement  of  Water  in  the  Plant  Tissues,  Influence  of  Heat 
and  Light  upon  Growth,  Irritability  of  Plants,  Classification,  etc.,  etc.  A  Model 
for  describing  a  Plant,  a  Series  of  Questions  for  Examination,  and  an  Index  and 
Glossary  are  also  given. 


LONGMANS,  GREEN,  &  CO.,  15  East  Sixteenth  Street,  New  York. 


L  ONGMA  NS,   GREEN,   &*  C  O. '  S  P  UBLICA  TIONS. 

PRACTICAL    ELEMENTARY    BIOLOGY. 

By  JOHN  BIDGOOD,  B.Sc.,  F.L.S.     With  226  Illustrations  and  full  Index. 

I2mo.     362  pages.     $1.50. 

"  The  types  described  in  the  following  pages  have  been  selected  as  fairly 
representing,  in  so  far  as  they  can  be  represented  in  a  book  of  this  size,  the 
vegetable  and  animal  worlds.  It  is  possible  that  a  better  selection  might  be 
made,  but  not  one,  I  think,  of  which  the  examples  can  be  more  conveniently 
procured,  cultivated,  and  examined,  while  retaining  their  capability  of  teaching 
so  much.  .  .  .  Many  of  the  illustrations  are  new  and  have  been  drawn  for 
this  book." — Author's  Preface. 

A   TEXT-BOOK    OF    ELEMENTARY    BIOLOGY. 

By  R.  J.  HARVEY  GIBSON,  M.A.,  F.R.S.E.  Demonstrator  of  Biology  in 
University  College,  Liverpool.  Illustrated  with  192  Engravings.  Crown 
8vo.  372  pages.  $1.75. 

In  this  book  the  author  has  kept  prominently  in  the  foreground  the  depend- 
ence of  Biology  on  Chemistry  and  Physics,  and  the  relationship  of  Morphologi- 
cal and  Physiological  Details  to  General  Principles.  Great  prominence  has 
been  given  to  the  Botanical  Aspect  of  Biology.  It  contains  chapters  on  Matter 
and  Energy,  Protoplasm,  Individual  and  Tribal  Life — Distribution  and  Classi- 
fication, The  Morphology  and  Physiology  of  the  Simplest  Living  Organisms — 
Protista,  Unicellular  Plants — Protophyta,  Unicellular  Animals— Protozoa, 
Metaphyta — Non-Vascularia,  Metaphyta — Vasculana,  Metazoa — Invertebrata, 
Metazoa— Vertebrata,  History  of  Biology,  etc. 

\* For  other  books  on  Botany,  Biology,  etc.,  see  Longmans,  Green,  «3°  Co. 's 
Catalogue  of  Educational  Works. 

ELEMENTARY    PHYSIOGRAPHY.      An    Introduc- 
tion to  the  Study  of  Nature. 

By  JOHN  THORNTON,  M.A.  With  10  Maps  and  161  Illustrations.  Crown 
8vo.  256  pages.  80  cents. 

This  volume  is  intended  to  serve  as  an  introduction  to  Science.  It  supplies 
such  a  knowledge  of  the  facts  and  laws  of  Nature  as  is  implied  in  the  expres- 
sive term  Physische  Erdkunde — an  acquaintance  with  the  physical  phenomena 
of  the  Earth.  It  contains  chapters  on  Matter  and  its  Properties,  Gravitation 
and  Specific  Gravity,  Cohesion  and  Chemical  Affinity,  Work  and  Energy, 
Chemical  Action  ;  Rocks,  their  Composition,  Classification,  and  Arrangement ; 
Interior  of  the  Earth,  Volcanoes,  etc.  ;  the  Sea,  the  Polar  Regions,  and  Ice  of 
the  Sea  ;  the  Atmosphere,  Evaporation  and  Condensation,  Dews,  Mist,  Fog, 
Rain,  and  Snow  ;  the  Sculpture  of  the  Land,  Weather,  and  Climate.  Changes 
in  the  Earth's  Surface  ;  Magnetism  and  Electricity  of  the  Earth,  Shape  and 
Movements  of  the  Earth,  etc. 

The  third  Edition  contains  a  short  account  of  recent  researches  on  Dew, 
and  gives  a  simple  explanation  of  Telescopes. 

ST.  PAUL'S  SCHOOL. 

"  I  have  been  using  Thornton's  Elementary  Physiography  for  two  years 
with  my  classes  beginning  the  study  of  science.  I  find  it  a  most  admirable 
book  and  can  certainly  recommend  it  from  a  personal  knowledge  of  it.  I 
shall  continue  its  use." — DR.  J.  MILNOR  COIT,  CONCORD,  N.  H. 


LONGMANS,  GREEN,  &  CO.,  15  East  Sixteenth  Street,  New  York. 


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WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
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